question 1

Let A = (1, 2, 3, 4) Let R be a relation on A^2 defined as

R = ((1; 1); (2; 2); (3; 3); (4; 4); (2; 3); (4; 3)). Determine if R is an equivalence relation.

question 2

Prove or disprove that ~ is an equivalence relation:

(a) Let N be the set of non-negative integers. A relation ~ on the set N as follows:

a ~ b if and only if a + b is an even integer.

(b) A relation ~ is defined on the set of integers as x ~ y <---> x = y:

(c) A relation ~ is defined on the set Z by x ~ y <---> x = ky ; k ∈ R.

(d) A relation ~ is defind on Z as x ~ y if and only if x <= y

(e) Let R+ be the set of positive real numbers. Let ~ be a relation on R+, dened as

a ~ b <---> a/b = 2^k, k ∈ Z