1. ## Relations

question 1
Let A = (1, 2, 3, 4) Let R be a relation on A^2 defined 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 defined on the set of integers as x ~ y <---> x = y:
(c) A relation ~ is defined 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+, dened as
a ~ b <---> a/b = 2^k, k ∈ Z

2. ## Re: Relations

Hey beast.

Hint: Equivalence relations have the property [a] = {b in X such that a ~ b}. Take a look at the wiki entry (it has a similar example to yours):

Equivalence relation - Wikipedia, the free encyclopedia

3. ## Re: Relations

Hey, chiro,

The equivalence class [a] is usually not even defined for non-equivalence relations.

4. ## Re: Relations

An equivalence relation must satisfy
1) reflexive: if a is in the set then (a, a) is in the relation.
2) symmetric: if (a, b) is in the relation, then (b, a) is also in the relation.
3) transitive: if (a, b) is in the relation and (b, c) is in the relation, then (a, c) is in the relation.

Are those true for these relations?