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Set Operators
Goals • Show how set identities are established • Introduce some important identities.
Copyright © Peter Cappello 2
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.
Copyright © Peter Cappello 3
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 = .
Copyright © Peter Cappello 4
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.
Copyright © Peter Cappello 5
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.
Copyright © Peter Cappello 6
Set IdentitiesIdentity Name of laws
A = AA U = A
Identity
A U = UA =
Domination
A A = AA A = A
Idempotent
Complement of A = A Complementation
A B = B AA B = B A
Commutative
Copyright © Peter Cappello 7
Identity Name of laws
A (B C)= (A B) CA (B C)= (A B) C
Associative
A (B C) = (A B) (A C) A (B C) = (A B) (A C)
Distributive
A B = A BA B = A B
De Morgan
A (A B) = AA (A B) = A
Absorption
A A = UA A =
Complement
Think like a mathematicianHow 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
Copyright © Peter Cappello 8
Copyright © Peter Cappello 9
Prove A B = A BVenn diagrams
1. Draw the Venn diagram of the LHS.
2. Draw the Venn diagram of the RHS.
3. Explain that the regions match.
Copyright © Peter Cappello 10
Prove A B = A BUse 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 )
Copyright © Peter Cappello 11
Prove A B = A BMembership Table
A B A B A B A B A BF 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
23 4A B
Let x be an arbitrary member of the Universe.In the table below, each column denotes the propositionfunction “x is a member of this set.”
Think like a mathematicianIs 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.Copyright © Peter Cappello 12
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
Copyright © Peter Cappello 13
Copyright © Peter Cappello 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?
