copyright 2004-2006 curt hill euler circles with venn diagrams thrown in for good measure
DESCRIPTION
Copyright 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 anotherTRANSCRIPT
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
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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
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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
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Fourth Example
p¬p
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Fifth Example
1p ¬q 2
q¬p 3qp
0¬q¬p
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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
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q p
qp
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¬( 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