The document covers key set operations such as union, intersection, difference, and complement, along with their definitions and examples. It details important set identities and laws, illustrating these concepts through diagrams and proofs. Furthermore, the text draws analogies between logic and set theory and discusses computer representation of sets using bit strings.

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
Identity
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

8. Think like a mathematician
How much is new here?
Logic Set
x  S S
False 
True Universe
 
 
 complement
 =
Can you mechanically produce
set identities from propositional
identities via this translation?
Example:
( x  A  x   )  x  A
A   = A
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Prove A  B = A  B
Venn diagrams
1. Draw the Venn diagram of the LHS.
2. Draw the Venn diagram of the RHS.
3. Explain that the regions match.

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Prove A  B = A  B
Use set operator definitions
1. A  B = { x | x A  B } (defn. of complement)
2. = { x | (x  A  B) } (defn. of  )
3. = { x | (x  A  x B) } (defn. of  )
4. = { x | (x  A  x  B) } (Propositional De
Morgan)
5. = { x | (x  A  x  B) } (defn. of complement )
6. = A  B (defn. of  )

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Prove A  B = A  B
Membership Table
A B A  B A  B A B A  B
F F F T T T T
F T T F T F F
T F T F F T F
T T T F F F F
1
2
3 4
A B
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.”

12. 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?
1. What is the set analog of p  q?
2. 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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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?