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M ACM 101 M ACM 101 Set Theory Page 1 M ACM 101 Set Theory Example Example - Direct Proof Prove that for all integers k and l, if k and l are both odd, then their product kl is also odd. Prove that if m is an even integer, then m + 7 is odd. Set Theory Example cont’d… - Contraposition Page 3 M ACM 101 Set Theory Example cont’d… - Contradiction Page 2 Page 4 M ACM 101 Set Theory Page 5 M ACM 101 Set Theory Page 6 Approach to Proofs Example If you are not given specific directions on how to do the proof, try the following: Prove that for all positive real numbers x and y, if the product xy exceeds 25, then x > 5 or y > 5. ! Try a direct proof If you suspect the theorem is invalid, try to find a counterexample ! ! If that fails, try an indirect proof: • • M ACM 101 Set Theory Page 7 M ACM 101 Set Theory Page 8 Chapter 3: Set Theory Definitions A set is a well-defined collection of objects. These objects are called elements and are said to be members of the set. If C, D are sets from a universe, we say that C is a subset of D and write C ! D, or D " C if every element of C is an element of D. We use capital letters such as A, B, C, … to represent sets and lowercase letters to represent elements. For a set A we write x ! A if x is an element of A. E.g. of sets A = {1,3,5,7,9} B = {x | x ! Z+, x is odd} C = {1,3,5,7,9} If, in addition, D contains an element that is not in C, then C is called a proper subset of D, and this is denoted by C # D or D $ C. The size, or cardinality, of a finite set A is the number of elements in the set, and is denoted | A | E.g. M ACM 101 Set Theory Page 9 M ACM 101 Set Theory Subsets Set Equality Note that for all sets C, D, C ! D if and only if "x [x ! C # x ! D]. For a given universe, the sets C and D are said to be equal, and we write C = D, when C ! D and D ! C. E.g. E.g. M ACM 101 Set Theory Page 11 M ACM 101 Set Theory Theorem Theorem - cont’d… For a given universe containing the sets A, B, C: (b) If A # B and B ! C, then A # C. (a) If A ! B and B ! C, then A ! C. Proof: Proof: Page 10 Page 12 M ACM 101 Set Theory Page 13 M ACM 101 Set Theory Theorem - cont’d… Definitions (c) If A ! B and B # C, then A # C. The null set, or empty set, is the set containing no elements and is denoted by % or {}. (d) If A # B and B # C, then A # C. E.g. For the set A, the power set of A, denoted P(A), is the set of all subsets of A. E.g. C = {1,2,3,4} Page 14