May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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Relations and Digraphs 2-13
Example 1 Students in a class are partitioned according to their surnames.
Example 2 A country is partitioned into states.
Example 3 A deck of playing cards is partitioned into four suits.
2.5 CONGRUENCE MODULO RELATION
Two integers a and b are said to be congruent modulo a positive integer m, if (a − b) is divisible
by m, that is, a − b = km, where k is an integer. It is written as a ≡ b (mod m).
THEOREM 2.2 The relation ‘congruence modulo m’ is an equivalence relation.
Proof:
1. The relation is reflexive as a − a = 0 is divisible by m, that is, a ≡ a (mod m).
2. The relation is symmetric because a ≅ b (mod m) ⇒
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