1. Copyright 2004-2006 Curt Hill Euler Circles With Venn Diagrams Thrown in for Good Measure

2. Copyright 2004-2006 Curt Hill Venn Diagrams Leonhard Euler (1707-1783) used them first They are more commonly associated with John Venn (1834-1923) Since Euler’s place in mathematical history is not in question, we will use Venn’s for the name

3. Copyright 2004-2006 Curt Hill Boolean Algebra and Set Theory are isomorphic This means that any theorem in one (and its proof) can be transformed into the other Variables in Boolean algebra convert to membership in a set Unions are Ors Intersections are Ands Complement is Negation All other operators in one have corresponding operators in another

4. Copyright 2004-2006 Curt Hill Venn Diagram Example

5. Copyright 2004-2006 Curt Hill Discussion The interior of the circle represents: –Members of the set –The variable true The exterior is: –Non-members of the set –The variable false The rectangle is –The universe of discourse –The variables being considered

6. Copyright 2004-2006 Curt Hill Second Example 0 21 3

7. Copyright 2004-2006 Curt Hill Discussion In two circles there are four areas 0 – not a member of either 1 – member of first but not the second 2 – member of the second but not the first 3 – member of both Of course, this numbering is completely arbitrary

8. Copyright 2004-2006 Curt Hill Another View There are also four ways to draw the circles –Overlapping –Two disjoint –Two identical circles –One circle contained in another These carry interpretation about the contents (or lack of contents) of areas 1-3 –This allows for some of the areas to be void

9. Copyright 2004-2006 Curt Hill Third Example 0 21 3 0 21 0 1 3 Disjoint, 3 is empty Contained, 2 is empty Normal, 4 areas

10. Copyright 2004-2006 Curt Hill Venn Diagram for Boolean Algebra One circle gives two areas –p–p –¬p If p is a constant true or false –One of areas is void

11. Copyright 2004-2006 Curt Hill Fourth Example p ¬p¬p

12. Copyright 2004-2006 Curt Hill Fifth Example 1 p  ¬q 2 q  ¬p 3 q  p 0 ¬q  ¬p

13. Copyright 2004-2006 Curt Hill Boolean interpretation All combinations of areas have a construction –3 – p  q –1,2,3 – p  q –0,2,3 – p  q –0,3 – p  q

14. Copyright 2004-2006 Curt Hill Diagram proofs Generate the diagrams for each side of an equivalence A tautology should have identical coloring –A contradiction should be different Venn diagrams provide a proof that is more graphic than truth tables –Yet less convincing than what we would like

15. Copyright 2004-2006 Curt Hill Prove p  ¬(q  p) The proof using Venn diagrams proceeds somewhat like that of a truth table Start with small pieces Build up from there Start with p  q

16. Copyright 2004-2006 Curt Hill q  p qp

17. Copyright 2004-2006 Curt Hill ¬( q  p) qp

18. Copyright 2004-2006 Curt Hill p  ¬( q  p) qp

19. Copyright 2004-2006 Curt Hill Another Proof Disprove –p  q ≡ q  p This is known as affirming the antecedent –Common logical fallacy An implication –If it is Thursday at 2 then I teach logic The fallacy –I am teaching logic, so it must be Thursday at 2.

20. Copyright 2004-2006 Curt Hill p  q qp q p q

21. Copyright 2004-2006 Curt Hill q  p qp p q p

22. Copyright 2004-2006 Curt Hill Do these look the same to you? p  q and q  p are not equivalent