1. Set Operators Goals Show how set identities are established
Introduce some important identities.

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Union
Let A & B be sets.
A union B, denoted A  B, is the set A  B = { x | x  A  x  B }.
Draw a Venn diagram to visualize this.
Example
O = { x  N | x is odd }.
S = { s  N | x  N s = x2 }.
Describe O  S.
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Intersection
Let A & B be sets.
A intersection B, denoted A  B, is the set A  B = { x | x  A  x  B }.
Draw a Venn diagram to visualize this.
Example
O = { x  N | x is odd }.
S = { s  N | x  N s = x2 }.
Describe O  S.
A & B are disjoint when A  B = .
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Difference
Let A & B be sets.
The difference of A & B, denoted A – B, is A – B = { x | x  A  x  B }.
Draw a Venn diagram to visualize this.
Example
O = { x  N | x is odd }.
S = { s  N | x s = x2 }.
Describe O – S.
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Complement
Let A be a set.
The complement of A is { x | x  A } = U – A.
Draw a Venn diagram to visualize this.
Example
O = { x  N | x is odd}.
Describe the complement of O.
Since I cannot overline in Powerpoint, I denote the complement of A as A.
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Set Identities
Identity
Name of laws
A   = A
A  U = A
A  U = U
A   = 
Domination
A  A = A
A  A = A
Idempotent
Complement of A = A
Complementation
A  B = B  A
A  B = B  A
Commutative
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Identity
Name of laws
A  (B  C)= (A  B)  C
A  (B  C)= (A  B)  C
Associative
A  (B  C) = (A  B)  (A  C)
A  (B  C) = (A  B)  (A  C)
Distributive
A  B = A  B
A  B = A  B
De Morgan
A  (A  B) = A
A  (A  B) = A
Absorption
A  A = U
A  A = 
Complement
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8. Think like a mathematician How much is new here?
Can you mechanically produce set identities from propositional identities via this translation? Example: ( x  A  x   )  x  A A   = A
Logic
Set
x  S S
False 
True
Universe     
complement  =
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9. Prove A  B = A  B Venn diagrams
Draw the Venn diagram of the LHS.
Draw the Venn diagram of the RHS.
Explain that the regions match.
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10. Prove A  B = A  B Use set operator definitions
A  B = { x | x A  B } (defn. of complement)
= { x | (x  A  B) } (defn. of  )
= { x | (x  A  x B) } (defn. of  )
= { x | (x  A  x  B) } (Propositional De Morgan)
= { x | (x  A  x  B) } (defn. of complement )
= A  B (defn. of  )
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11. Prove A  B = A  B Membership Table
Let x be an arbitrary member of the Universe.
In the table below, each column denotes the proposition
function “x is a member of this set.” A B
A  B
A  B F T A B 3 4 2 1
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Think like a mathematician Is membership table the analog of truth table?
With 3 propositional variables, a truth table has 23 rows.
With 3 sets, do we have 23 regions?
Does this generalize to n sets?
What is the analog of modus ponens?
What is the set analog of p  q?
What is the set analog of a tautology?
If interested, see chapter 12 of textbook.
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13. Analogy between logic & sets
In logic: p  q ≡ p  q
Its set analog is P  Q
Set analog of modus ponens
( p  ( p  q ) )  q is
Complement[ P  ( P  Q ) ]  Q
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14. Computer Representation of Sets
There are many ways to represent sets.
Which is best depends on the particular sets & operations.
Bit string: Let | U | = n, where n is not “too” large: U = { a1, …, an }.
Represent set A as an n-bit string.
If ( ai  A )
bit i = 1;
else
bit i = 0.
Operations ,  , _ are performed bitwise.
In Java, Set is the name of an interface.
Consider a Java set class (e.g., BitStringSet), where | U | is a constructor parameter.
What data structures might be useful to implement the interface?
What public methods might you want?
How would you implement them?
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