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\begin{document}
\title{U3\_AOS1\_Topic00\_Mathematical notation}
\author[1]{Yanik}%
\affil[1]{ConnectApp}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
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\section*{Question Group information}\label{question-group-information}
\textbf{Name:~}Mathematical notation
\textbf{Summary:}
Before we head out on the wonderful road trip that is Methods, let's
take a glance at the arsenal of tools that'll make our journey that much
more enjoyable. We're talking about notation! Mathematics is a language
in itself --- it's the expression of logic using symbols, numbers and
operations. If we can get familiar with the symbols component of maths,
the entire road trip will feel much smoother. So what we're going to
cover here is all of the relevant notation and conventions that we'll
need not only for the different topics, but for the exam too.
\textbf{Videos:}
\textbf{~ ~ Video1 url:}
\textbf{~ ~ Video 1 title:}
\textbf{~ ~ Video 1 thumb url:}
\textbf{Start at:}
\textbf{End at:}
\par\null
\pagebreak
\textbf{Tutorial number: 1}
\textbf{Prompt:}~How do we represent sets of numbers?
\textbf{Title:} Notations for sets and their meanings
There are a few types of notations for sets we need to know, and the following tables break it down for us.
\section*{Sets of numbers:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-sets-of-numbers/mathematical-notation-sets-of-numbers}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_sets\_of\_numbers.png
{\label{418636}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-order-of-numbers-sets/mathematical-notation-order-of-numbers-sets}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_order\_of\_number\_sets.png
~
{\label{365931}}%
}}
\end{center}
\end{figure}
We can summarise everything from the table and the Venn diagram as follows:
\begin{enumerate}
\item Real numbers are the most expansive set of numbers in Methods.
\item The real number set contains the rational numbers, which are the real numbers that can be expressed as a fraction.
\item The rational number set contains the integer numbers, which are the fractions in rational numbers that simplify into whole numbers.
\item The integer number set contains the natural numbers, which are the whole numbers that start from $1$, $2$, $3$, etc., up to positive infinity.
\end{enumerate}
This leaves us with the irrational numbers ($Q'$), which from the Venn diagram is the largest oval ($R$ --- real numbers) excluding the yellow oval ($Q$ --- rational numbers).
\section*{Notation for sets:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-types-of-notation-for-sets/mathematical-notation-types-of-notation-for-sets}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_types\_of\_notation\_for\_sets.png
{\label{491048}}%
}}
\end{center}
\end{figure}
\textbf{}
Let's explore some more examples of how we can use the different types of notations.
Say we want to represent the following number line using some notation.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-interval-example-1/mathematical-notation-interval-example-1}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_interval\_example\_1.png
{\label{912909}}%
}}
\end{center}
\end{figure}
The easiest way to represent this is using interval notation, though we will explain the other relevant way to write this down.
\begin{enumerate}
\item $[-1, 2]$
\begin{itemize}
\item[\textbullet] This simply uses interval notation to represent the values that $x$ can take.
\item[\textbullet] This is read as ``the numbers from $-1$ inclusive to $2$ inclusive''
\item[\textbullet] Note that the end points of the interval are \textbf{closed circles}, meaning that we must \textbf{include} each end point.
\end{itemize}
\item $-1 \leq x \leq 2$
\begin{itemize}
\item[\textbullet] This is another simple method to represent the numbers $x$ can take.
\item[\textbullet] This approach formally says ``$x$ is greater than or equal to $-1$, though also less than or equal to $2$''.
\item[\textbullet] In everyday language, this means ``$x$ lies between $-1$ inclusive and $2$ inclusive''.
\end{itemize}
\end{enumerate}
\newline
\newline
What if we change the example so that the interval looks something like this?\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-interval-example-2/mathematical-notation-interval-example-2}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_interval\_example\_2.png
~
{\label{616768}}%
}}
\end{center}
\end{figure}
Again, let's have a look at the different methods we can use to represent this interval.
\begin{enumerate}
\item $(-1, 2]$
\begin{itemize}
\item[\textbullet] The main difference in this example compared to the previous example is the open circle at $-1$. The round, exclusive bracket in this interval notation accounts for this.
\item[\textbullet] This is read as ``the numbers from $-1$ \textbf{exclusive} to $2$ inclusive".
\end{itemize}
\item $-1 < x \leq 2$
\begin{itemize}
\item[\textbullet] Again, this is very similar to the previous example, though the different inequality accounts for the open circle.
\item[\textbullet] This is read as ``$x$ lies between $-1$ \textbf{exclusive} and $2$ inclusive".
\end{itemize}
\end{enumerate}
\newline
\newline
\subsection*{Tip!}
A common error made in VCE Methods is the mix up between the square and round bracket notations, and which one represents inclusive and exclusive intervals. There are many ways that individual students remember which is which, though here is one way that could help us.
If we write down a round bracket and a square bracket like ``\; $($ \;" and ``\; $[$ \;", we can use a bit of creativity to see that the round bracket seems to take up less ``space" inside the symbol than the square bracket. Specifically, the round bracket curves around the space on the inside, while the square bracket almost makes a box around that space.
Using this, we can say that the square brackets ``\; $[$ \;" and ``\; $]$ \;" are \textit{inclusive} because they take up or \textbf{include} more space inside the symbol. On the other hand, we can say that the round brackets ``\; $($ \;" and ``\; $)$ \;" are \textit{exclusive} because they take up less space, or \textbf{exclude} more space, inside the symbol.
\newline
\newline
Let's look at a slightly different example.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-interval-example-3/mathematical-notation-interval-example-3}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_interval\_example\_3.png
{\label{881257}}%
}}
\end{center}
\end{figure}
Here, we don't have a line joining the numbers... instead we're just including certain numbers. The main way of representing this set of numbers is as follows.
\begin{enumerate}
\item $\{-1, 0, 1, 2\}$
\begin{itemize}
\item[\textbullet] Formally, this is read as ``the set of numbers $-1$, $0$, $1$ and $2$".
\item[\textbullet] In everyday language, this just means ``the numbers $-1$, $0$, $1$ or $2$".
\end{itemize}
\end{enumerate}
Notice how we had to switch the notation for this set? The set of numbers is no longer an interval, or a continuous line of numbers, rather it's a bunch of individual numbers. As soon as this comes up, we need to switch to our general set notation using curly brackets, and use commas to just list the numbers that are included.
\newline
\newline
Lastly, what if we combine them into this weird looking example?\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-interval-example-4/mathematical-notation-interval-example-4}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_interval\_example\_4.png
{\label{611220}}%
}}
\end{center}
\end{figure}
We have a couple of individual numbers, but also a continuous interval of numbers. Here's what it should look like:
\[x \in \{-1\} \cup [\,0, 1) \cup \{2\}\]
Looks weird, hey? We'll explain some of these extra symbols in the next couple of tutorials!
\pagebreak
\textbf{Tutorial number: 2}
\textbf{Prompt:}~How can we use sets to give meaning to our maths?
\textbf{Title:} Incorporating symbols with sets
\par\null
Here, we're going to look at how sets and symbols are used together in maths. Sometimes our functions are limited to certain values, or our graphs may be restricted to a particular domain. These symbols allow us to express this in the most convenient way. The following tables look into these in detail.
\section*{Element notation:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-elements-notation/mathematical-notation-elements-notation}
\caption{{\{\{s3\_url\}\}/tutorial-02/mathematical\_notation\_elements\_notation.png
~
{\label{853210}}%
}}
\end{center}
\end{figure}
\section*{Subset notation:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-subset-notation/mathematical-notation-subset-notation}
\caption{{\{\{s3\_url\}\}/tutorial-02/mathematical\_notation\_subset\_notation.png
~
{\label{903847}}%
}}
\end{center}
\end{figure}
The symbols under element and subset notation are crucial, as they let us explain how sets are related to our variables, such as $x$ and $y$, and with other sets.
Keep in mind that numbers as well as variables, like $x$, are considered as elements, meaning that whenever we use notation, we must use the element symbols ``$\in$" and ``$\notin$".
\newline
\newline
\section*{Set connectors:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-set-connectors1/mathematical-notation-set-connectors1}
\caption{{\{\{s3\_url\}\}/tutorial-02/mathematical\_notation\_set\_connectors.png
~
{\label{580940}}%
}}
\end{center}
\end{figure}
Typically, the union and intersection symbols are used in probability, which will be covered in the probability topics. We do use the union symbol, however, when linking or joining individual sets to create a larger set, as we can see in the table's example.
\section*{Other notation for sets:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-other-notation-for-sets1/mathematical-notation-other-notation-for-sets1}
\caption{{\{\{s3\_url\}\}/tutorial-02/mathematical\_notation\_other\_notation\_for\_sets.png
~
{\label{916908}}%
}}
\end{center}
\end{figure}
It's just a bunch of symbols and signs at the moment, so let's put it into context. Let's go back to the final example of the first tutorial.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-interval-example-41/mathematical-notation-interval-example-41}
\caption{{\{\{s3\_url\}\}/tutorial-01/mathematical\_notation\_interval\_example\_4.png
{\label{113805}}%
}}
\end{center}
\end{figure}
As we said before, here's how we'd represent it using our notation:
\[x \in \{-1\} \cup [\,0, 1) \cup \{2\}\]
Let's build this up one step at a time.
\begin{enumerate}
\item ``$x \in$" \newline First of all we use ``$x \in$" to begin our set.
\begin{itemize}
\item[\textbullet] Since $x$ represents a value, we consider it an element --- not a \textit{set} of numbers --- and thus use the ``element of" sign ($\in$).
\end{itemize}
\item ``$x \in \{-1\}$" \newline Always working from smallest to largest (or left to right), the first number in our set is just the number $-1$.
\begin{itemize}
\item[\textbullet] Because this is just the one number by itself, we need to use our set notation with the \{ \} brackets.
\item[\textbullet] Hence, the first component of our set is purely the number $-1$ written in set notation as $\{-1\}$.
\end{itemize}
\item ``$x \in \{-1\} \cup [0, 1)$" \newline Now notice that our interval contains disjointed intervals and individual numbers.
\begin{itemize}
\item[\textbullet] As soon as we have disjointed, or disconnected intervals, we need to use our set connector tool: $\cup$.
\item[\textbullet] Thus, we join the next interval, written as $[0, 1)$, onto our set so far using the $\cup$ union symbol, jotting it down as $\{-1\} \cup [0, 1)$.
\end{itemize}
\item ``$x \in \{-1\} \cup [0, 1) \cup \{2\}$" \newline Finally, we only have one more separate number left to deal with.
\begin{itemize}
\item[\textbullet] Again with individual numbers, we'll use our set notation to write it as $\{2\}$.
\item[\textbullet] To add it onto our set so far we'll chuck in the union symbol in between, and we'll end up with the same set as above.
\end{enumerate}
\pagebreak
\textbf{Tutorial number: 3}
\textbf{Prompt:} What symbols can we use to translate everyday language
into maths and thereby make our working out quicker?
\textbf{Title:}~Writing mathematical statements using sets and symbols
~
Let's suppose we want to find the value of $y$ when $x = 1$ in the following equation:
\[y = 4x + 1\]
The method may be simple enough, so we'll take a quick look.
\begin{align*}
y &= 4x + 1 \newline
&Sub \; x = 1 \newline
y &= 4(1) + 1 \newline
y &= 4 + 1 \newline
y &= 5
\end{align*}
What if we wanted to take this further? Perhaps we want to write down a conclusion of what we just found using mathematical language?
To do this, let's look at some of the common notation that can spruce up our working out.
\section*{Statements:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-statement-starters/mathematical-notation-statement-starters}
\caption{{\{\{s3\_url\}\}/tutorial-03/mathematical\_notation\_statement\_starters.png
~ ~
{\label{597984}}%
}}
\end{center}
\end{figure}
\section*{Notation for functions and probability:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-functions-and-probability/mathematical-notation-functions-and-probability}
\caption{{\{\{s3\_url\}\}/tutorial-03/mathematical\_notation\_functions\_and\_probability.png
~ ~ ~
{\label{389314}}%
}}
\end{center}
\end{figure}
\section*{Geometry-related notation:}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-geometry/mathematical-notation-geometry}
\caption{{\{\{s3\_url\}\}/tutorial-03/mathematical\_notation\_geometry.png ~ ~ ~
{\label{152412}}%
}}
\end{center}
\end{figure}
Now that we've come across these symbols and their meanings, let's have a go at using them on our first example in this tutorial.
Again, let's suppose we want to find the value of $y$ when $x = 1$.
\begin{align*}
y &= 4x + 1 \newline
&Sub \; x = 1 \newline
\Rightarrow y &= 4(1) + 1 \newline
y &= 4 + 1 \newline
\Rightarrow y &= 5 \newline
\newline
&\therefore y = 5 \; when \; x = 1
\end{align*}
It looks beautiful, doesn't it? For now, we'll leave it at this --- when the need for them crops up in the rest of our Methods journey, we'll use and explain each piece of notation so that you understand where and how each symbol can be used. When we know how to use these symbols in the right places we can do wonderful things in the language of mathematics!
\pagebreak
\par\null
\textbf{Tutorial number: 4}
\textbf{Prompt:}~What are the different styles of writing set notation?
\textbf{Title:}~Set notation and set builder notation
Set notation can be used in a variety of ways. One of its uses is to represent individual numbers that make up a set. We can take this further by using it to construct more flexible sets. This is called set builder notation.
Set builder notation allows us to represent more than individual numbers --- we can even use this notation to write down intervals of numbers just like interval notation! We'll explain the key components of set builder notation here.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-set-builder-notation/mathematical-notation-set-builder-notation}
\caption{{\{\{s3\_url\}\}/tutorial-04/mathematical\_notation\_set\_builder\_notation.png
~ ~ ~
{\label{195476}}%
}}
\end{center}
\end{figure}
Let's examine what this means with a few examples.
\section*{Example 1:}
Let's say we had the following interval defined for us:
\[x \in (0, 1]\]
We'll translate this set (which is in interval notation) to its equivalent in set builder notation.
\[\{x: 0 < x \leq 1\}\]
If we were to read this in everyday language, it would go something like this:
\begin{center}
``The set of $x$ such that $x$ lies between $0$ exclusive and $1$ inclusive."
\end{center}
\section*{Example 2:}
We can still do so much more with this notation. Let's revisit the first example in this tutorial and see what it would look like using set builder notation. Here's the original set.
\[x \in \{1,2,3,4,5\}\]
Translating this into set builder notation would give us something like this.
\[\{x \in N: 1 \leq x \leq 5\}\]
If we read this in everyday language, it would sound like this:
\begin{center}
``The set of the natural numbers $x$ such that $x$ lies between $1$ and $5$ inclusive."
\end{center}
Notice how we still use inequalities to represent a range of numbers? This range of numbers isn't a continuous range, meaning that it \textbf{doesn't include} every single number between $1$ and $5$, such as $1.00001$ and $4.\dot{3}$. The fact that we defined $x$ as a natural number at the very beginning means that $x$ can only take positive whole numbers. Then we specified that $x$ lied between $1$ and $5$ inclusive, which then means that it can only take the numbers $1$, $2$, $3$, $4$ and $5$ as they are the only positive whole numbers in that range.
\subsection*{Extension:}
We could've also written the set as follows.
\[\{x \in Z: 1 \leq x \leq 5\}\]
This set means the exact same thing as the previous, even though we defined $x$ as being an integer number instead of a natural number. The only integer numbers between $1$ and $5$ inclusive are still $1$, $2$, $3$, $4$ and $5$.
Note: in Example 1, we did not include any ``$x \in R$" at the beginning, and we still assumed that it was all real numbers in a continuous interval. The assumption is that \textbf{if there is no specific set of numbers defined}, such as $x \in Q$ or $x \in R$, \textbf{then we assume it is part of the real number set}.
\section*{Example 3:}
What if we had disjointed intervals in a single set? Let's convert the following interval notation into set builder notation.
\[x \in [0, 1] \cup [2, \infty)\]
Translating it would give us this.
\[\{x: 0 \leq x \leq 1, x \geq 2\}\]
This is read as:
\begin{center}
``The set of $x$ such that $x$ lies between $0$ and $1$ inclusive, or $x$ is at least $2$."
\end{center}
Alternatively, we could write it in two separate sets and then join them together, just like with the interval notation.
\[\{x: 0 \leq x \leq 1\} \cup \{x: x \geq 2\}\]
Both are interpreted the same way and mean the same thing. This just goes to show how much flexibility set builder notation has.
\newline
\newline
VCAA has used a variety of notations throughout their years of examinations, and while set builder notation occurs the least, it in fact catches and trips up many students for not knowing its meanings.
\pagebreak
\section*{Question 1:~}\label{question-1}
\textbf{Title:~}~Write down a set in interval notation.
\textbf{Tags:}~Interval notation, number lines
\textbf{Question prompt:}
Examine the following number line and the set of numbers it represents.
\par\null\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-mc11/mathematical-notation-mc11}
\caption{{\{\{s3\_url\}\}/mc-01/mathematical\_notation\_mc01.png ~ ~ ~ ~
{\label{875145}}%
}}
\end{center}
\end{figure}
The interval notation that correctly defines the set of numbers
represented by the number line is:~
\par\null
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
\par\null
$x \in (-\infty, -2) \cap [-1, 1] \cap (2, \infty)$
~ ~ Explanation:
This isn't right because of the symbols you've used.
When using interval notation, we need to be weary of a couple of things: the type of brackets (whether they are inclusive or exclusive); and the type of symbol used to connect separate chunks of numbers together. In this answer, we haven't done either of these two things quite right.
\section*{Type of brackets:}
Don't forget that on the number line, closed circles mean inclusive and open circles mean exclusive.
By choosing this answer, you may have gotten mixed up between what the square and round brackets mean. It is important to remember that square brackets are used for \textbf{inclusive} numbers, while round brackets are used for \textbf{exclusive} numbers, not the other way around. Since there are closed circles at $-2$ and $2$, we need to use square brackets at these points, rather than round brackets; there are also open circles at $-1$ and $1$, meaning we need to use round brackets at these numbers, rather than square brackets.
Keep in mind that we \textbf{always} use round brackets for positive and negative infinity. One way to remember this is that the infinity symbol is curved/round, and so we should always use a round bracket next to it.
\section*{Connector symbol:}
With interval notation, a general rule is that we almost always use the union ``$\cup$" symbol to connect multiple sets. What we're actually doing by using the intersection ``$\cap$" symbol is telling the examiner that we want all the \textit{common} points in all three sets. In other words, the $\cap$ symbol tells us to find all the numbers that occur in each of the three intervals. If we look at each set carefully, there aren't any numbers that are part of all three intervals, meaning that a simplified version of this answer is
\[(-\infty, -2) \cap [-1, 1] \cap (2, \infty) = \varnothing\]
which makes our answer look like
\[x \in \varnothing.\]
What this means is that $x$ can take no number at all, which clearly doesn't make sense if we look at the number line and the set of numbers that it represents.
\newline
\newline
\par\null
B:
\par\null
$x \in (-\infty, -2) \cup [-1, 1] \cup (2, \infty)$
~ ~ Explanation:
This isn't right - your brackets are letting you down!
Remember that when using interval notation, we need to be weary of a couple of things: the type of brackets (whether they are inclusive or exclusive); and the type of symbol to connect separate chunks of numbers together. In this answer, we've done the second requirement correctly, though not the first.
\section*{Type of brackets:}
Don't forget that closed circles mean inclusive, and open circles mean exclusive.
By choosing this answer, you may have gotten mixed up between what the square and round brackets mean. It is important to remember that square brackets are used for \textbf{inclusive} numbers, while round brackets are used for \textbf{exclusive} numbers, not the other way around. Since there are closed circles at $-2$ and $2$, we need to use square brackets at these points, rather than round brackets; there are also open circles at $-1$ and $1$, meaning we need to use round brackets at these numbers, rather than square brackets.
Keep in mind that we \textbf{always} use round brackets for positive and negative infinity. One way to remember this is that the infinity symbol is curved/round, and so we should always use a round bracket next to it.
\section*{Connector symbol:}
With interval notation, a general rule is that we almost always use the union ``$\cup$" symbol to connect multiple sets. Here, we've used the correct $\cup$ symbol in the correct spot --- just in between each interval we want to connect.
\newline
\newline
\par\null
C:
\par\null
$x \in (-\infty, -2] \cap (-1, 1) \cap [2, \infty)$
~ ~ Explanation:
\par\null
This is so close to being right - your connecting symbols are letting you down!
Remember that when using interval notation, we need to be weary of a couple of things: the type of brackets (whether they are inclusive or exclusive); and the type of symbol to connect separate chunks of numbers together. In this answer, we've done the first requirement correctly, though not the second.
\section*{Type of brackets:}
Well done! You correctly recalled that closed circles mean inclusive, and open circles mean exclusive. We can say that the set is \textbf{inclusive} of $-2$ and $2$, though \textbf{exclusive} of $-1$ and $1$. Therefore, we use square brackets where we write $-2$ and $2$, and round brackets where we write $-1$ and $1$.
\section*{Connector symbol:}
With interval notation, a general rule is that we almost always use the union ``$\cup$" symbol to connect multiple sets. What we're actually doing by using the intersection ``$\cap$" symbol is telling the examiner that we want all the common points in all three sets. In other words, the $\cap$ symbol tells us to find all the numbers that occur in each of the three intervals. If we look at each set carefully, there aren't any numbers that are part of all three intervals, meaning that a simplified version of this answer is
\[(-\infty, -2] \cap (-1, 1) \cap [2, \infty) = \varnothing\]
which makes our answer look like
\[x \in \varnothing.\]
What this means is that $x$ can take no number at all, which clearly doesn't make sense if we look at the number line and the set of numbers that it represents.
\newline
\newline
\par\null
D: (correct)
\par\null
$x \in (-\infty, -2] \cup (-1, 1) \cup [2, \infty)$
~ ~ Explanation:
Yes! This is correct.
Let's take a look at how we approach this type of question.
First, we're given a number line with various values included. We're asked to write the set using interval notation, so we should recall that there are two things that contribute to writing this kind of notation: the type of brackets (whether they are inclusive or exclusive); and the type of symbol to connect separate chunks of numbers together.
\section*{Type of brackets:}
When figuring out what brackets to use, we should look at the circles on the number line to see if the endpoints are included or not. Recall that closed circles mean inclusive, and open circles mean exclusive. We can say that the set is \textbf{inclusive} of $-2$ and $2$, though \textbf{exclusive} of $-1$ and $1$. Therefore, we use square brackets where we write $-2$ and $2$, and round brackets where we write $-1$ and $1$.
Keep in mind that we \textbf{always} use round brackets for positive and negative infinity. One way to remember this is that the infinity symbol is curved/round, and so we should always use a round bracket next to it.
\section*{Connector symbol:}
Since we have multiple sets, we also need to choose a symbol to connect them. A general rule is that we almost always use the union ``$\cup$" symbol to connect multiple sets. Here, we've used the correct $\cup$ symbol in the correct spot --- just in between each interval we want to connect.
\par\null
\section*{Question 1 Hint Menu}\label{question-1-hint-menu}
Copy/paste the following tutorials:
\begin{itemize}
\tightlist
\item
\emph{} How do we represent sets of numbers?
\item
How can we use sets to give meaning to our maths?
\end{itemize}
\textbf{Question analysis}
\par\null
We're given a number line with distinct portions. Note that there is a mixture of open and closed circles, meaning that we need to think about what notation we use to represent this.
Taking a look at this number line, one of its biggest characteristics is that it's made up of intervals of numbers. This should raise a question: what kind of notation do we use for writing intervals?
Secondly, have a look at the structure of the intervals --- they're disjointed or separate intervals. What methods have we learned to deal with connecting different intervals together?
\par\null\par\null
\pagebreak
\section*{Question 2:~}\label{question-2}
\textbf{Title:} Identify the set that represents a number line.
\textbf{Tags:}~Set notation, interval notation, number lines
\textbf{Question prompt:}
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\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mathematical-notation-mc2/mathematical-notation-mc2}
\caption{{\{\{s3\_url\}\}/mc-02/mathematical\_notation\_mc02.png
{\label{660864}}%
}}
\end{center}
\end{figure}
All of the following sets, except for one, correctly represent the number line above.
The set of numbers that does \textbf{not} correctly define the number line above is:
\par\null
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
\par\null
A:
\par\null
$\{x: x \leq 2, x \neq -1\}$
~ ~ Explanation:
No, this statement is actually correct.
This is one of the correct methods of representing the number line of $x$. While interval notation is a very user-friendly method in writing continuous lines of numbers, we can still write them in the same way using set builder notation. Here, a \textbf{top-down approach} is used to construct the set. This means that we define a large set of numbers, then exclude certain portions out of it. The key characteristic making this answer top-down is that the set starts off with an inequality, then removes a part of that section out.
The basics of set builder notation involve the variable name at the front, followed by a colon. Since the variable is $x$, given by the number line, we can already start our set like this
\[\{x:\ldots\}\]
where the dots will be filled in using the top-down approach.
\section*{1. Starting at the top:}
Here, we will define a large set of values that all of our number line is contained by. We can see that the line extends from negative infinity all the way up to $2$ inclusive, given by the closed circle. We can also notice an open circle along the way at $-1$, meaning that $-1$ must be excluded from the set which we'll deal with later.
We can write $x \leq 2$, and this would cover the entirety of the red number line. This will make up the large set of numbers that we mentioned earlier. Hence, we should get something like this so far.
\[\{x: x \leq 2\}\]
So far so good, though we still need to account for the $-1$ --- we've started at the top, and now we need to work our way down to the details.
\section*{2. Working our way down:}
From the open circle on the number line, we can tell that $x$ cannot be $-1$. If we translate this to maths, we should get a statement that looks like this: $x \neq -1$. Now we have a statement that allows us to exclude $-1$ from the set, though the only issue is how to incorporate it into our set.
Reflecting back to arsenal of symbols and notation, you may remember the comma symbol. Its meaning is specifically ``where", though it can be used to separate different statements from each other. This means that we can use a comma to separate our two statements, $x \leq 2$ and $x \neq -1$, from each other in our set. Thus, we should get something that looks like this.
\[\{x: x \leq 2, x \neq -1\}\]
And this gives us a correct way to represent the number line using set notation!
\par\null
B:
\par\null
$\{x: x \leq 2\} \cup \{x: x \neq -1\}$
~ ~ Explanation:
No, this statement is actually correct.
This is one of the correct methods in representing the number line of $x$. While interval notation is a very user-friendly method in writing continuous lines of numbers, we can still write them in the same way using set builder notation. Here, a \textbf{bottom-up approach} is used to construct the set. This means that the set is built from the ground up using sections of the entire set. The key characteristic making this set botom-up is that the it is made of individual blocks.
\section*{1. Starting at the bottom:}
The goal here is to identify the different sections and characteristics of the number line. There are a couple of ways we can dissect this example, though in line with the notation in this option, there are two main characteristics:
\begin{enumerate}
\item $x \leq 2$
\begin{itemize}
\item[\textbullet] If we were to write this in set builder notation by itself, it would look like this \newline \newline
$\{x: x \leq 2\}$
\end{itemize}
\newline
\item $x \neq -1$
\begin{itemize}
\item[\textbullet] If we were to write this in set builder notation by itself, it would look like this \newline \newline
$\{x: x \neq -1\}$
\end{itemize}
\end{enumerate}
\newline
The issue here is how we combine these characteristics into an overall set.
\section*{2. Working our way up:}
We have two individual sets defined so far, and we need to link them together. Recalling our set connector notation, we can use the union ``$\cup$" symbol to join sets together into one. We can put the $\cup$ symbol between our two sets to achieve this, and hence we obtain a final set that correctly defines the number line:
\[\{x: x \leq 2\} \cup \{x: x \neq -1\}\]
\par\null
C: (correct)
$x \in (-\infty, 2] \, \backslash -1$
~ ~ Explanation:
Yes! This statement is incorrect.
The reason why this option incorrectly defines the number line is due to the lack of proper use of notation. The issue here is that $-1$ should not be written by itself --- it requires set notation around it. With one minor tweak to this statement, it would be a correct representation of the number line.
Remember that whenever we are writing sets of numbers, we must include every number we write as part of some form of set or interval notation. When it comes to a continuous line of numbers, we use one of the two ways of interval notation, inclusive square brackets and exclusive round brackets. When it comes to individual numbers, we need to use set notation with curly brackets.
This option attempts to write the set in a \textbf{top-down approach}. This means that we define a large set of numbers then exclude certain portions out of it, hence the use of the backslash (excluding sign). The characteristic that makes this top-down is that we start off with the entire set of numbers below $2$, then remove the $-1$ from it.
\section*{1. Starting at the top:}
The statement correctly defines the large set of numbers that it's dealing with, ranging from negative infinity to 2 inclusive and represented by the $(-\infty, 2]$. Doing this, we have started from the top by writing a set that includes more numbers that we need. From here, we can see an open circle at $-1$, meaning that we must somehow remove $-1$ from our large interval. This is where we starting working down.
\section*{2. Working our way down:}
Now we need to exclude $-1$ from our set, which is an example of working our way down. From our arsenal of tools we can recollect that the symbol to exclude specific things from a set is the backslash symbol. Since we use the backslash symbol in between a set and what we want to exclude, we have to place it next to our current set of numbers like this.
\[x \in (-\infty, 2] \, \backslash\]
After the backslash we write down what we want to exclude. We know that we want to exclude the number $-1$, though we can't write $-1$ next to the backslash by itself. Remember, this is still a set of numbers meaning we need to use either interval or set notation, even to represent individual numbers. So to write down the number $-1$, we actually have to write it in \textbf{set notation} as $\{-1\}$.
Combining this with the notation we have so far, and the fact that this is a number line of $x$, we get the following.
\[x \in (-\infty, 2] \, \backslash \{-1\}\]
This would be a correct method in representing the set of numbers on the number line. Notice the difference between the statement in this option and the statement we've arrived at? Anything represented in a set of number must be written using notation of some kind, either interval or set notation.
\par\null
D:~
\par\null
$(-\infty, -1) \cup (-1, 2]$
~ ~ Explanation:
No, this statement is actually correct.
This is a correct method to represent the number line, and it uses a \textbf{bottom-up approach} to do so. This means that the set is built from the ground up using sections of the entire set. Here's the explanation. The characteristic that makes this bottom-up is that we break the number line into two different sections, then join them together.
\section*{1. Starting at the bottom:}
This first step involves breaking the number line down to its individual components. The key thing to look out for are breaks in the number line, or where the number line splits or separates. These tell us where a section ends and where another starts.
Let's try to find the first section. Since this is a continuous number line, the easiest method is interval notation. Interval notation works smallest to largest, or left to right, so let's start by looking at the smallest number.
The number line seems to extend all the way to negative infinity, given by the red arrow pointing to the left. If we trace the line towards the right, the first obstacle we come across is an \textbf{open circle} at $-1$. Don't forget that open circles mean exclusion in interval notation. Hence, we've come across our first break in the number line, and if we express what we have so far as an interval, we get something that looks like this.
\[(-\infty, -1)\]
And that's the first section done, though we still have some of the number line left to trace. If we start tracing from $-1$ again, which is where the next section starts, and continue towards the right, we come across the closed circle end point at $2$. A closed circle at $2$ means that we include the number, while the open circle at $-1$ where we started means that we exclude the number. Thus, the next section should look something like this.
\[(-1, 2]\]
And that finishes up the entire number line! That's the first step done, let's move to the second.
\section*{2. Working our way up:}
With all our sections identified and written down, we just need a way to join them up and reach a final answer. Keep in mind the different tools we have from our set connectors. The one we want is the union ``$\cup$" symbol; this allows us to link different sets into one overall set. Hence, putting this symbol in between our two sections, we should get something that looks like this.
\[x \in (-\infty, -1) \cup (-1, 2]\]
Note that this is still a number line of $x$, meaning it must be $x$ that is an element of the sets, which is why there is an ``$x \in$" at the front.
\par\null\par\null
\section*{Question 2 Hint Menu}\label{question-2-hint-menu}
Copy/paste the following tutorials:
\begin{itemize}
\tightlist
\item
\emph{} How do we represent sets of numbers?
\item
How can we use sets to give meaning to our maths?
\end{itemize}
\textbf{Question analysis}
\par\null
This number line seems to have a small chunk missing from it. Keeping this in mind, we could use a process of elimination to work out the answer. Ask yourself which options correctly show numbers ranging across the entire red line, as well as the number $-1$ being excluded from the set. Then, the final answer that you're left with should be the correct answer.
\par\null
\pagebreak
\section*{Question 3:~}\label{question-3}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
~
\par\null\par\null
\pagebreak
\section*{Question 4:~}\label{question-4}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null
~
\pagebreak
\section*{Question 5:~}\label{question-5}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null
\pagebreak
\section*{Question 6:~}\label{question-6}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null\par\null
\pagebreak
\section*{Question 7:~}\label{question-7}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null
\pagebreak
\section*{Question 8:~}\label{question-8}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null
\pagebreak
\section*{Question 9:~}\label{question-9}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null
\pagebreak
\section*{Question 10:~}\label{question-10}
\textbf{Title:~}Calculate/identify/deduce
\textbf{Tags:}
\textbf{Multiple choice options:~}(write `correct' in brackets after the
correct option)
A:
~ ~ Explanation:
\par\null
B:
~ ~ Explanation:
\par\null
C:
~ ~ Explanation:
\par\null
D:
~ ~ Explanation:
\par\null
\pagebreak
\subsection*{Question 1}\label{question-1}
\textbf{Summary Title:~}
\textbf{Tag}:
\textbf{Source}: Connect
\textbf{Context:~}
\par\null
\textbf{Question Part (a):}
\textbf{Prompt:~}
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
\textbf{Question Part (b):}
\textbf{Prompt:~}~
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
\textbf{Question Part (c):}
\textbf{Prompt:~}~
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
~\textbf{Question Part (d):}
\textbf{Prompt:~}~
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
\pagebreak
\subsection*{Question 2}\label{question-2}
\textbf{Summary Title:~}
\textbf{Tag}:
\textbf{Source}: Connect
\textbf{Context:~}
\par\null
\textbf{Question Part (a):}
\textbf{Prompt:~}
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
\textbf{Question Part (b):}
\textbf{Prompt:~}~
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
\textbf{Question Part (c):}
\textbf{Prompt:~}~
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
\par\null
~\textbf{Question Part (d):}
\textbf{Prompt:~}~
\textbf{Display type:~}Short answer-box
\textbf{No. marks:~}
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