- Is R reflexive?
- Is R symmetric?
- Is R antisymmetric?
- Is R transitive?
- Is R a function?
Answer:
- Yes. The relation will be reflexive because \(x=y\) is a condition. Therefore \(\langle x, y \rangle\), where \(x=y\), will always be in the relation set.
- Yes. The relation is symmetric because the order of parameters does not matter. \(\langle 1,2 \rangle\) and \(\langle 2,1 \rangle\) will both be true.
- No. The relation is not antisymmetric because \(\langle \sqrt{0.2}, \sqrt{0.8} \rangle\) and \(\langle \sqrt{0.8}, \sqrt{0.2} \rangle\) are both elements, but \(\sqrt{0.2} \neq \sqrt{0.8}\).
- No. The relation is not transitive because \(\langle 1,0 \rangle\) is an element, \(\langle 0,-1 \rangle\) is an element, but \(\langle 1,-1 \rangle\) is not.
- No. The relation is not a function because \(\sqrt{0.2}\) is true for both \(\langle \sqrt{0.2}, \sqrt{0.2} \rangle\) and \(\langle \sqrt{0.2}, \sqrt{0.8} \rangle\).