1. ???????,??????? Please state the following Principles and Definitions (using symbolic pattern): (a) Equivalence Relations. (b) Composition of Relations. (c) A Linearly Ordered Set.
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Reflexivity: ∀x ∈ A, xRx Symmetry: ∀x, y ∈ A, if xRy then yRx Transitivity: ∀x, y, z ∈ A, if xRy and yRz then xRz Show more…
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Let A = {a, b, c, d, e} and the equivalence relation on the set A is defined as follows: R = {(a, a), (a, e), (b, b), (b, c), (c, b), (c, c), (d, d), (e, a), (e, e)}. Which of the following form a partition of the set A? a) {{a, e}, {b, c}, {d}} b) {a, b, c, d, e} c) {{a, c, e}, {b, d}} d) {a, b, c, d}, {e}
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Let R be a relation on a set U. State the definitions of the following: (a) R is reflexive. (b) R is symmetric. (c) R is antisymmetric. (d) R is transitive. (e) R is an equivalence relation. (f) R is a partial ordering.
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6. List the ordered pairs in the equivalence relations produced by the partition {a}, {b, c}, {d, e, f} of the set A = {a, b, c, d, e, f}. 7. Given a number p > 1, let (x, y) ∈ R iff x mod p = y mod p. Prove that R is an equivalence relation. What are the equivalence classes of R?
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