1. Membership Tables: Proving Set Identities with One Example

2. 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

3. 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

4. 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.

5. Venn Diagram VerificationX Z Y

6. 4 8

7. Venn Diagrams for more than 3 sets
Source:

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

9. “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