copyright 2004-2006 curt hill euler circles with venn diagrams thrown in for good measure

Copyright 2004-2006 Curt Hill

Euler CirclesWith Venn Diagrams Thrown

in for Good Measure


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


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


Venn Diagram Example


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


Second Example

0

21

3


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


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


Third Example0

213

0 21

0 1

3

Disjoint, 3 is empty

Contained, 2 is empty

Normal, 4 areas


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


Fourth Example

p¬p


Fifth Example

1p ¬q 2

q¬p 3qp

0¬q¬p


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


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


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


q p

qp


¬( q p)

qp


p ¬( q p)

qp


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.


p q

qp

q

p q


q p

qpp qp


Do these look the same to you?

p q and q p are not equivalent

copyright 2004-2006 curt hill euler circles with venn diagrams thrown in for good measure

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