Question

Which of the following is an equivalence relation defined on set $A=\left\{1,2,3\right\}$

• A. $\left\{(1,1),(1,2),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\right\}$
• B. $\left\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2)\right\}$
• C. $\left\{(1,1),(2,2),(3,1),(1,3),(3,3)\right\}$
• D. $\left\{(1,2),(2,1),(1,3),(3,1)\right\}$

Answer 1

The correct option is C

$\left\{(1,1),(2,2),(3,1),(1,3),(3,3)\right\}$

Explanation for the correct option:

Option (C): $\left\{(1,1),(2,2),(3,1),(1,3),(3,3)\right\}$

Any set is said to be equivalence if it is reflexive, symmetric, and transitive.

For reflexive- if $\left\{1,2,3\right\}\in A$ then $(1,1)\in A,(2,2)\in A,(3,3)\in A$

For symmetric-if $(1,3)\in A$ then $(3,1)\in A$

For transitive-if $(1,3)\in A$ and $(3,1)\in A$ then $(1,1)\in A$

Explanation for the incorrect options:

Option $\left(A\right)$ $\left\{(1,1),(1,2),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\right\}$ because the given ordered pair doesn't match the conditions of an equivalence relation.

$(2,3)\in A,(3,1)\in A,(2,1)\notin A$

Option $\left(B\right)$ $\left\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2)\right\}$ because the given ordered pair doesn't match the conditions of an equivalence relation.$(3,3)\notin A$

Option $\left(D\right)$$\left\{(1,2),(2,1),(1,3),(3,1)\right\}$ because the given ordered pair doesn't match the conditions of an equivalence relation.

$(3,3)\notin A$

Hence, option $\left(C\right)$ is the correct answer.