1. Is R reflexive?
  2. Is R symmetric?
  3. Is R antisymmetric?
  4. Is R transitive?
  5. Is R a function?

Answer:

  1. 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. 
  2. 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.
  3. 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 \(\sqrt{0.2} \neq \sqrt{0.8}\).
  4. 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.
  5. No. The relation is not a function because \(\sqrt{0.2}\)  is true for both \(\langle \sqrt{0.2}, \sqrt{0.2} \rangle\) and  \(\langle \sqrt{0.2}, \sqrt{0.8} \rangle\).