copyright 2004-2006 curt hill euler circles with venn diagrams thrown in for good measure
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Copyright 2004-2006 Curt Hill
Euler CirclesWith Venn Diagrams Thrown
in for Good Measure
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
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
Copyright 2004-2006 Curt Hill
Venn Diagram Example
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
Copyright 2004-2006 Curt Hill
Second Example
0
21
3
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
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
Copyright 2004-2006 Curt Hill
Third Example0
213
0 21
0 1
3
Disjoint, 3 is empty
Contained, 2 is empty
Normal, 4 areas
Copyright 2004-2006 Curt Hill
Venn Diagram for Boolean Algebra
• One circle gives two areas– p– ¬p
• If p is a constant true or false– One of areas is void
Copyright 2004-2006 Curt Hill
Fourth Example
p¬p
Copyright 2004-2006 Curt Hill
Fifth Example
1p ¬q 2
q¬p 3qp
0¬q¬p
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
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
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
Copyright 2004-2006 Curt Hill
q p
qp
Copyright 2004-2006 Curt Hill
¬( q p)
qp
Copyright 2004-2006 Curt Hill
p ¬( q p)
qp
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.
Copyright 2004-2006 Curt Hill
p q
qp
q
p q
Copyright 2004-2006 Curt Hill
q p
qpp qp
Copyright 2004-2006 Curt Hill
Do these look the same to you?
p q and q p are not equivalent
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