Q: Let $S = \{1, 2, 3\}$. Each of the following subsets of $S \times S$ gives a relation on $S$.
Select all those that give equivalence relations on $S$.
a) $\{(1, 1),(1, 2),(2, 1),(2, 2),(3, 2),(3, 3)\}$
(b) $\{(1, 1)\}$
(c) $\{(1, 1),(1, 2),(1, 3),(2, 1),(2, 2),(2, 3),(3, 1),(3, 2)\}$
(d) $\{(1, 1),(1, 3),(2, 2),(2, 3),(3, 1),(3, 2),(3, 3)\}$
I'm stuck on this question. I said it was b) but I'm not sure if I'm right.
I would really appreciate your help. Thanks!
Answer: none(may be you had done typo while typing options)
a relation which is identity, symmetric and transitive is called the equivalence relation!
Clearly, $(b)$ is not equivalence relation! Since it is not identity relation(clearly you can see $(2,2), (3,3)∉\{(1,1)\}$)
Similarly, $(c)$ is not equivalence relation(as $(3,3)∉$Set)
further, (d) is not transitive! (Since $(2,1)∉$Set) hence not equivalence!
Infact, (a) is also not equivalence! Since it is not symmetric (as $(2,3)∉$set)