The binary relation E on real numbers set is defined as foll

The binary relation E on real numbers set is defined as follows: inner product a, b inner product E equivalent a - b is an integer Prove that E is an equivalence relation. Explain what is the equivalence class [5]_E.

Solution

Proof.

I. Reflexive: Suppose a E. Then a a = 0, which is an integer. Thus, aEa.

II. Symmetric: Suppose a, b E and aEb. Then a b is an integer. Since
b a = (a b), b a is also an integer. Thus, bRa.

III. Suppose a, b R, aRb and bRc. Then a b and b c are integers. Thus,
the sum (a b) + (b c) = a c is also an integer, and so aRc.

Thus, E is an equivalence relation on R.

ii).

Let R be an equivalence relation on A and let a A. The set [a] = {x|aRx}
is called the equivalence class of a.

[5]E = { 5,6,7,............}