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Published byPrincess Streat Modified over 2 years ago

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Membership Tables: Proving Set Identities with One Example

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Proof techniques we teach in Discrete Mathematics Proof techniques we teach in Discrete Mathematics Direct Proofs Direct Proofs Proofs by Contradiction Proofs by Contradiction Proofs by Contrapositive Proofs by Contrapositive Proofs by Cases Proofs by Cases Mathematical Induction (Strong Form?) Mathematical Induction (Strong Form?) Proof techniques we do NOT teach in Discrete Mathematics Proof techniques we do NOT teach in Discrete Mathematics Proof by one example Proof by one example Proof by two examples Proof by two examples Proof by a few examples Proof by a few examples Proof by many examples Proof by many examples

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Two sets X and Y are equal if X and Y have the same elements. Two sets X and Y are equal if X and Y have the same elements. To prove sets X and Y are equal, prove if To prove sets X and Y are equal, prove if then, and if, then

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Example: Determine the truth value of Proof: Let. Then x is a member of X, or x is a member of. Hence, x is a member of X, or x is a member of Y and x is a member of Z. If x is a member of X, then x is a member of and x is a member of. Otherwise, x is a member of Y and Z and hence a member of and. Either way, x is a member of and. Therefore,.. Now show the other direction.

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Venn Diagram Verification X Z Y

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4 2 1 3 6 5 7 8

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Source: http://www.combinatorics.org/Surveys/ds5/VennGraphEJC.html Venn Diagrams for more than 3 sets 4 sets5 sets

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XYZ 11111111 11001111 10101111 10001111 01111111 01000100 00100010 00000000 Membership Tables 1 2 3 4 5 6 7 8

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Never underestimate the value of a good example. --Tom Hern, Bowling Green State University Invited Address at 2005 Fall Meeting of the Ohio Section of MAA Discrete Mathematics textbooks containing descriptions of membership tables: 1) Kenneth H. Rosen, Discrete Mathematics and Its Applications, 5 th Ed., McGraw Hill, 2003, p. 91. 2) Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, 5 th Ed., Pearson, 2004, pp. 143-144.

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