Letthe set of natural numbers, define a relation on is eventhena)R is an equivalence relationb)these are only two class of with respect to this relationc)d)Correct answer is option 'A,C'. Can you explain this answer?

Question

Answer

If option B said this....
There are only two distinct equivalence classes with respect to this relation.
then it would definitely be correct.

Answer

Can anyone tell me how option 2 is incorrect.

Answer

Let, a+b =2k
b+c = 2m
a+b+b+c = 2k+2m
a+c = 2(k+m-b) which is also even, so it is transitive .
a+ a = 2a , so it is reflexive.
a+b =b+a ,so it is symmetric.
so it is equivalence relation.

Now, cl(2) means the number related to 2 so the number added with 2 gives a even number ,
cl(2) is the set of all even natural numbers.
other side, cl(1) means the number added with 1 ,gives an even number,
cl(1) means all the odd natural numbers
so intersection of cl(1) and cl (2) is phi.

Answer

A and c are correct answer by definition of equivalence relation option a is correct and the intersection is 0

Answer

Yes there are two eqivalence classes only

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