Question

Consider the set A={a,b}, The smallest equivalence relation that can be defined on A is

a) { } b){(a,a),(b,b)} c){(a,b),(b,a)} d) AXA

Answer 1

Answer:

for reflexive (a,a)(b,b) required.

for symmetric (a,b)(b,a) required.

for transitive(a,b)(b,c)(c,a) required. but in set c is not given so it is transitive

so the smallest equivalence relation that can be defined is {(a,a)(b,b)(a,b)(b,a)}

Answer 2

The smallest equivalence relation that can be defined on A is $\{(a,a),(b,b)\}$and option (b) is correct.

Step-by-step explanation:

Given:

The set $A=\{a,b\}.$

To Find:

The smallest equivalence relation that can be defined on A.

Solution:

As given-the set $A=\{a,b\}.$

A relation is an equivalence relation if and only if it is reflexive, symmetric and transitive.

The smallest equivalence relation on the set $A=\{a,b\} is:$

$Relation \ R=\{(a,a),(b,b)\}.$

$\{a,a\} \{b,b\}\in R \rightarrow Reflexive$

$\{a,a\}\{b,b\} \in R \rightarrow Symmetric$

$\{a,a\}\{b,b\} \in R \rightarrow Transitive$

Thus,the smallest equivalence relation that can be defined on A is $\{(a,a),(b,b)\}$and option (b) is correct.

PROJECT CODE#SPJ3