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Use Theorem 7.2 to write a computer program (or to develop an algorithm) for the recognition of equivalence relations on a finite set.
Theorem 7.2
Given a set A with |A| = n and a relation R on A, let M denote the relation matrix for R. Then
(a) R is reflexive if and only if In ≤ M.
(b) R is symmetric if and only if M = Mtr.
(c) R is transitive if and only if M ∙ M = M2 ≤ M.
(d) R is antisymmetric if and only if M ⋂ Mtr ≤ In. (The matrix M ⋂ Mtr is formed by operating on corresponding entries in M and Mtr according to the rules 0 ⋂ 0 = 0 ⋂ 1 = 1 ⋂ 0 = 0 and 1 ⋂ 1 = 1 - that is, the usual multiplication for 0 s and /or 1 s.) |
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