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1. (25 points) For each of the five following binary relations R on the set A, determine whether
it has each of the four following properties, and briefly justify your answer. Properties: i. R is reflexive. ii. R is symmetric.
iii. R is antisymmetric.
iv. R is transitive. Relations: (a) A is the positive integers and R is the set of pairs (m, n) such that every prime number
that divides am. also divides n and every prime number that divides 'n. also divides m. (b) A is all subsets of the natural numbers, that is, A = ‘P(N), and (S, T) E R if and only if
S' Q T. (c) A is the set of all finite sequences of lower case letters of the English alphabet (which
are not necessarily words, for example: app, apple, zzyzz) and R is the set of all pairs
(at) such that s aé t and a would appear before If in a dictionary. So (app, apple) 6 R
but (zzyzz, zyz) 9! R. (d) A is all subsets of the nonnegative integers, that is, A = 39(N), and R is the set of pairs
(3,5?) such that there exists a bijection f : S —> T. (Hint: you may assume that if
f : S —> T is a bijection, then its inverse function f ‘1 : T —> S is also a bijection, and
also that the composition of two bijections is a bijection.) (e) A is the set N of all nonnegative integers and R is the set of pairs (m,n) such that
m2 > n.

Top Answer

a) Reflexive, Symmetric, Transitive, but not... View the full answer

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