Discrete Mathematics with Graph Theory (3rd Edition) 69

# Discrete Mathematics with Graph Theory (3rd Edition) 69 -...

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Section 3.3 67 Antisymmetric: If A, B E P(S), A ~ Band B ~ A, then we have IAI ~ IBI and IBI ~ IAI, so IAI = IBI. Since P(S) does not contain different sets of the same cardinality, it follows that A = B. Transitive: Suppose A ~ Band B ~ C. If A = 0,then IAI = ° ~ 101 no matter what C is, so we'd have A ~ C. If A = S, then A ~ B means B = Sand B ~ C means C = S, so A = B = C = S and A ~ C. 9. I: A x B ---+ B x A defined by I(a, b) = (b, a) is a one-to-one onto function. 10. (a) [BB] False. Let X = {I}, Y = {2}, Z = {3}. Then ((1,2),3) E (X x Y) x Z but ((1,2),3) fj. X x (Y x Z) (a set whose second coordinates are ordered pairs). (b) [BB] Define I: (X x Y) x Z ---+ X x (Y x Z) by I((x, y), z) = (x, (y, z)). 11. (a) Reflexivity: For any set A, "A is a one-to-one onto function A ---+ A, so A has the same cardinality as itself. Symmetry: If A and B have the same cardinality, then there is a one-to-one onto function I: A ---+ B. Such a function has an inverse 1-1 : B ---+ A which is one-to-one and onto (because it has an inverse), so B and A have the same cardinality. Transitivity: Suppose A, B and C are sets such that A and B have the same cardinality and B and C have the same cardinality. Then there is a one-to-one onto function
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