Let AB be sets let fAB f A B and let S be an equivalence rel

Let A,B be sets, let f:AB f A B and let S be an equivalence relation on B. Let R be the relation on A given by xRy if, and only if f(x)Sf(y) x R y if, and only if f x S f y . Show that R is an equivalence relation on A.

Solution

1. Check reflexive property

Let x be any element in A

So we need to check if xRx

f(x)S f(x) because S is equivalence

Hence, xRx . HEnce, R is reflexive since x is arbitrary element in A

2. Check if R is symmetric

Let, xRy

Hence, f(x)Sf(y)

S is equivalence ehnce, S is symmetric hence, f(y)Sf(x)

Hence, yRx

Hence, R is symmetric

3. Check for transitive property

Let, xRy,yRz

Hence, f(x)Sf(y),f(y)Sf(z)

S is equivalence hence transitive so ,f(x)Sf(z)

Hence, xRz

Hence, R is transitive and hence equivalence relation on A