Question

Let X be non-empty set. A set P = {A, B, C...] of non-empty subsets of X will
be called a partition of X if​

Answer 1

LetX={1,2,3}

P(X)=Power set of X=Set of all subsets of X.

={ϕ,{1}},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}

Since{1}⊂{1,2}

{1}R{1,2}

∵ofABCD,all elements of Aarein B

ARB means A⊂B

here,relation is R={(A,B):AandBaresets,ACB}

Since every set is a subset of itself.

ACA∴(A,A)ϵR,R is reflexive.

To check whether symmetric or not,

If(A,B)ϵ,then(B,A)ϵR.

If(A,B)ϵR,A⊂B

butB⊂Aisnottrue.

eg:−LetA={1}andB={1,2}

As all elements of A are in BA⊂B.

But all elements of B are not in A

∴B⊂A is not true.

∴R is not symmetric.

Since (A,B)ϵRand(B,C)ϵRofA⊂BandB⊂C

then A⊂B

⇒(A,C)ϵR

So,If(A,B)ϵ Rand(B,C)ϵR,

then(A,C)ϵR

∴R is transitive.

hence R is reflexive and transitive but not symmetric.

hence, R is not an equivalence relation since it is not symmetric