In 19—31, (1) prove that the relation is an equivalence relation, and (2) describe the distinct equivalence classes of each relation.

20. E is the relation defined on Z as follows:
For every m ,   n ∈ Z ,   m   E   n ⇔ 4 | ( m − n ) .

To prove:

(1) The given relation is an equivalence relation,

(2) Describe the distinct equivalence classes of each relation.

Given information:

E is the relation defined onZ as follows: For all m, n ∈Z ,

m E n ⇔ 4 | ( m - n).

Proof:

(1). A relation will be equivalence relation if it is reflexive, symmetric and transitive.

To proof: E is an equivalent relation

A=ZE={(m,n)∈A×A|4|( m−n)}

Reflexive:

Let x∈A=Z

x−x=0=0⋅4

By the definition of divides, 4 divides x − x and thus (x,x)∈E.

Since (x,x)∈E for all x∈A and thus E is indeed reflexive.

Symmetric:

Let us assume that (x,y)∈E. by the definition of E :

4|(x−y)

By the definition of divides, there exists an integer k such that:

x−y=4k

Multiply each side by − 1:

−(x−y)=−(4k)

Use the distributive property:

y−x=4(−k)

By the definition of divides, 4 divides y − x as − k is an integer (because k is an integer).

(y,x)∈E

Since (x,y)∈E implies (y,x)∈E,E is symmetric.

Transitive:

Let us assume that (x,y)∈E and (y,z)∈E