Question

For natural numbers x and y, define xRy if and only if x^2 + y is even. Prove that R is an equivalence relation on the set of natural numbers and find the quotient set determined by R. What would the quotient set be? can this proof be explained in detail?

Answer 1

x^2+x is even for all natural numbers as x and x^2 have teh same parity

So R is reflexive

Let x^2+y be even then, x^2 and y are of the same parity ie both are odd or both are even

x^2 has same parity as x hence x and y are of the same parity hence, x and y^2 are of the same parity

Hence, x+y^2 is even

Hence, R is symmetric

Let, x^2+y be even and y^2+z be even

Hence, x and y are of same parity and y,z are of the same parity

Hence, x,z are of the same parity

Hence, x+z^2 is even

Hence , R is transitive and hence an equivalence relation

As we see above xRy if and only if x and y are of the same parity

So we have two equivalence classes

[2]=set of all even natural numbers

[1]=set of all odd natural numbers

Quotient set

N/R={[2],[1]} is two element set