(a) What are the properties of an equivalence relation? (Defi ne them, don't just name them.)
(b) Show that parity (i.e. whether a number is odd or even) is an equivalence relation.
(a) Relation xRy defined on the set X × X is an equivalence relation if it is:
Reflexive. For all x ϵ X we have xRx.
Symmetric. For any x, y ϵ X, xRy implies yRx .
Transitive. If xRy and yRz, then xRz.
(b) Define the parity relation mPn on the set Z × Z: mPn if and only if m - n is an even number.
The relation is reflexive because n - n = 0 (an even number).
The relation is symmetric because if mPn, then m - n is an even number, that it, m - n = 2k (k ϵ Z) and n - m = 2(-k). Finally, 2(-k) is an even number and nPm.
The relation is transitive. Let mPn and nPq hold. Then m - n = 2k and n - q = 2l (k, l ϵ Z).
We conclude that m - q = (m - n) + (n - q) = 2(k + l) is an even number, too. Therefore, mPq.
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