Membership Tables: Proving Set Identities with One Example

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

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 then , and if , then

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.

Venn Diagram Verification X Z Y

4 2 1 3 6 5 7 8

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

Membership Tables 1 2 3 4 5 6 7 8 X Y Z 1

“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, 5th Ed., McGraw Hill, 2003, p. 91. 2) Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, 5th Ed., Pearson, 2004, pp. 143-144.