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 .
(1) The given relation is an equivalence relation,
(2) Describe the distinct equivalence classes of each relation.
E is the relation defined onZ as follows: For all m, n ∈Z ,
m E n ⇔ 4 | ( m - n).
(1). A relation will be equivalence relation if it is reflexive, symmetric and transitive.
To proof: E is an equivalent relation
By the definition of divides, 4 divides x − x and thus .
Since for all and thus E is indeed reflexive.
Let us assume that by the definition of E :
By the definition of divides, there exists an integer k such that:
Multiply each side by − 1:
Use the distributive property:
By the definition of divides, 4 divides y − x as − k is an integer (because k is an integer).
Since is symmetric.
Let us assume that
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