MiscellaneousSummation notationTake the sum \(x_1+x_2+x_3+...+x_n\)We can express this sum using the summation symbol \(\Sigma\): \(\sum_{i=1}^n x_i=x_1+x_2+x_3+...+x_n\)What does the notation mean? What does the superscript mean? The subscript?IMPORTANT: Whenever I do not specify superscript and subscript, always assume \(i=1, n=n\)How do we interpret the following? \(\Sigma_{i=30}^{35}N_i\)Do example 1, p56Rules of sums & Newton's binomial formulaAdditivity property: \(\Sigma(a_i+b_i)=\Sigma a_i + \Sigma b_i\)Homogeneity property: \(\Sigma (c\cdot a_i)=c\Sigma a_i\), where \(c\) is a constantFrom the homogeneity property, it follows that \(\Sigma c=n\cdot c\). Why?Do example 2 p60: Derive the fact that the sum of the difference between \(x_i\) and its arithmetic mean \(\mu_x\) is equal to zero: \(\Sigma (x_i-\mu_x)=0\)Useful formulasGauss' formula: \(\sum i= \frac{1}{2} n(n+1)\)Proof:\(x=1+2+...+(n-1)+n\)We can rewrite this as \(x=n+(n-1)+...+2+1\)From these we can write \(2x=[n+(n-1)+...+2+1]+[1+2+...+(n-1)+n]\)Rearranging we get \(2x=(n+1)+(n-1+2)+...+(2+n-1)+(1+n)=(n+1)+(n+1)+...+(n+1)+(n+1)\)Finally, we get \(2x=n\left(n+1\right)\). Solving for \(x\): \(x=\frac{1}{2}n(n+1)\). QEDTwo other useful formulas are \(\sum i^2=\frac{1}{6}n(n+1)(2n+1)\) and \(\sum i^3=[\sum i]^2\)Newton's binomial formula: \((a+b)^m= a^m+\left( \begin{array}{c} m \\ 1 \end{array} \right)a^{m-1}b+...+\left( \begin{array}{c} m \\ m-1 \end{array} \right)ab^{m-1}+\left( \begin{array}{c} m \\ m \end{array} \right)b^m\) where the binomial coefficients \(\left( \begin{array}{c} m \\ k \end{array} \right)=\frac{m(m-1)...(m-k+1)}{k!}\) are defined for \(m=1, 2, ...\) and \(k=0,1,2,...,m\)Application of Newton's formula:\((a+b)^3=a^3+\left( \begin{array}{c} 3 \\ 1 \end{array} \right)a^{2}b+\left( \begin{array}{c} 3 \\ 2 \end{array} \right)ab^{2}+b^3\). We can use the formula from above to write \((a+b)^3=a^3+\frac{3}{1}a^{2}b+ \frac{3\cdot 2}{1\cdot 2}ab^{2}+b^3=a^3+3a^2b+3ab^2+b^3\)Double sumsIt's possible to calculate the sum of sums by using the following formula: \(\sum_{i=1}^m \sum_{j=1}^na_{ij}\)Do example 1 p65: \(\sum_{i=1}^3\sum_{j=1}^4(i+2j)=\sum_{i=1}^3[(i+2)+(i+4)+(i+6)+(i+8)]=\sum_{i=1}^3(4i+20)=(4+20)+(8+20)+(12+20)=84\)A few aspects of logicDo example 1, p66Propositions: assertions that are either true or false. When an assertion contains one or more variables for which it can be true or false we say it is an open propositionThe implication arrow: \(\Longrightarrow\)\(P\Longrightarrow Q\) you read as "p implies q" or "if p then q"The logical equivalence arrow: \( \Longleftrightarrow\)\(P \Longleftrightarrow Q\) you read as "p if and only if q"Do example 2 p67Necessary and sufficient conditions: If \(P \Longrightarrow Q\) we say that p is sufficient condition for qIf \(P\Longrightarrow Q\) we say that q is necessary condition for pExplainSolving equationsDo examples 3-4 pp68-69Mathematical proofsEvery mathematical theorem can be formulated as an implication. Indeed, all proofs of mathematical theorems rely on the establishment of implications between premises (or assumptions) and conclusions.Examples of proof methods:Direct proof: We start from the premises and we keep deriving their implications until we get to the conclusions.Indirect proof: We deny the conclusions and show that the premises must also be false. The indirect proof relies on the fact that \(P\Longrightarrow Q\) is equivalent to \(\backsim Q \Longrightarrow \backsim P\) (non-q implies non-p)Do example 1 p72Essentials of set theoryWe partially covered this last weekYou should be familiar with the following notions/notationsSet and elements: \(S= \{e_1, e_2, e_3, ..., e_n\}\)Two sets, A and B, are said to be equal if every element in A is also an element in BDo examplesProperty of a setWe can use the following notation to specify the property of a set: \(S=\{e: p\}\), where e=typical elements and p= defining propertiesExample: the budget set. \(B=\{(x, y): px+py\leq m, x\geq 0, y\geq 0\}\). Explain.Set membershipWe covered this in class last week. Remember the meaning of the following symbols:\(\subset\)\(\subseteq\)\(\in\)\(\notin\)Set operationsUnion: \(\cup \). All the elements that belong to at least one of the sets.Intersection: \(\cap\). All the elements that belong to both sets.Minus: \(\setminus\) .All the elements that belong to one set but not the other.Do example 1 p76Other important notions:Disjoint set: the empty set. For example, the intersection between two sets that do not share any membersThe universal set, \(\Omega\): The set containing all potential subsets of a family of sets.Complement set: \(A^c=\Omega\setminus A\). It contains all elements of the universal set not contained in A.