On the set $A=\{1,2,3,4,5\}$, type three such relations $a,b,c$ so that $a = a^{-1}, b = b^{-1}, c = c^{-1}$

Question

On the set $A=\{1,2,3,4,5\}$, type three such relations $a,b,c$ so that $a = a^{-1}, b = b^{-1}, c = c^{-1}$. How do we write such relations on the set?

Answer 1

A relation on a set $A$ ($A$ to $A$) is a subset of $A\times A$.

Let $R$ be a relation from $A$ to $A$. Then the inverse of $R$ can be defined by $R^{-1}=\{(b, a)|(a, b)∈R\}$

First find $A \times A$ which is equal to $\{(1,1), (1,2), (1,3), (1,4), \cdots, (5,4), (5,5)\}$ then try to find subsets of this set to satisfy the condition.