The relation RZ times Z is defined as for a b elementof Z a

The relation R:Z times Z is defined as for a, b elementof Z, (a, b) elementof R if a + b is even. Prove all the properties: reflexive, symmetric, anti-symmetric, transitive that relation R has. If relation R does not have a property, explain why. Is R an equivalence relation?

Solution

Ans-

(a,b)R(a,b)R if and only if 3a+5b3a+5b is divisible by 88

Prove that RR is an equivalence relation.

Attempt:

Reflexive:

a~a if and only if 3a+5a3a+5a is divisible by 8

Since 3a+5a=8a3a+5a=8a is divisible by 8, the relationship is reflexive.

Symmetry:

Let a,b,kZa,b,kZ

aRbaRb = 3a+5b=8k3a+5b=8k

bRabRa = 5b+3a=8k5b+3a=8k

Therefore, it\'s symmetric

Transitivity:

Let a,b,c,i,kZa,b,c,i,kZ

aRbaRb = 3a+5b=8i3a+5b=8i

bRcbRc =