8-Sets-2.ppt

CSC102 - Discrete Structures
(Discrete Mathematics)
Set Operations

Set operations: Union
Formal definition for the union of two sets:
A U B = { x | x  A or x  B } or
A U B = { x  U| x  A or x  B }
Further examples
 {1, 2, 3}  {3, 4, 5} = {1, 2, 3, 4, 5}
 {a, b}  {3, 4} = {a, b, 3, 4}
 {1, 2}   = {1, 2}
Properties of the union operation
 A   = A Identity law
 A  U = U Domination law
 A  A = A Idempotent law
 A  B = B  A Commutative law
 A  (B  C) = (A  B)  C Associative law
2

Set operations: Intersection
Formal definition for the intersection of two sets:
A ∩ B = { x | x  A and x  B }
Examples
 {1, 2, 3} ∩ {3, 4, 5} = {3}
 {a, b} ∩ {3, 4} = 
 {1, 2} ∩  = 
Properties of the intersection operation
 A ∩ U = A Identity law
 A ∩  =  Domination law
 A ∩ A = A Idempotent law
 A ∩ B = B ∩ A Commutative law
 A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law
5

Disjoint sets
Formal definition for disjoint sets:
Two sets are disjoint if their intersection is the
empty set
Further examples
 {1, 2, 3} and {3, 4, 5} are not disjoint
 {a, b} and {3, 4} are disjoint
 {1, 2} and  are disjoint
• Their intersection is the empty set
  and  are disjoint!
• Because their intersection is the empty set
7

Exercise
• What is the cardinality of AUB where A and B
are two finite sets?
• |A ∪ B| = |A| + |B| ???
• Incorrect
• |A ∪ B| = |A| + |B| − |A ∩ B|???
• Correct
8

Set operations: Difference
Formal definition for the difference of two sets:
A - B = { x | x  A and x  B }
Sometimes denoted by AB.
Further examples
 {1, 2, 3} - {3, 4, 5} = {1, 2}
 {a, b} - {3, 4} = {a, b}
 {1, 2} -  = {1, 2}
The difference of any set S with the empty set will be the set S
9

Exercise
• What is the cardinality of A∩B where A and B
are two finite sets?
• |A∩B | = |B| - |B - A|???
• Correct
• |A∩B | = |A| - |A - B|???
• Correct
• |A∩B | = |A| + |B| - (|A - B| + |B - A|)???
• Incorrect
10

Complement sets
Formal definition for the complement of a set:
A = { x | x  A } = Ac
 Or U – A, where U is the universal set
Further examples (assuming U = Z)
 {1, 2, 3}c = { …, -2, -1, 0, 4, 5, 6, … }
 {a, b}c = Z
Properties of complement sets
 (Ac)c = A Complementation law
 A  Ac = U Complement law
 A ∩ Ac =  Complement law
11

How to prove a set identity
For example: A ∩ B = B - (B - A)
Four methods:
1. Using the basic set identities
2. Proving that each set is a subset of each other
3. Using set builder notation and logical
equivalences
4. Using membership tables (we’ll not study it)
• Similar to truth tables
15

First Proof
A ∩ B = B - (B - A)
A B
A∩B
B-A
B-(B-A)
16

1. Proof by Set Identities
A  B = A - (A - B) = B – (B – A)
Proof: A - (A - B)
= A - (A  Bc) A - B = A  Bc
= A  (A  Bc)c Same as above
= A  (Ac  B) De Morgan’s Law
= (A  Ac)  (A  B) Distributive Law
=   (A  B) Complement Law
= A  B Identity Law
17

2. Showing each is a subset of the others
Second Example Proof
 (A  B)c = Ac  Bc
Proof: Want to prove that
(A  B)c  Ac  Bc and Ac  Bc  (A  B)c
(i) x  (A  B)c
 x  (A  B)
  (x  A  B)
  (x  A  x  B)
  (x  A)   (x  B)
 x  A  x  B
 x  Ac  x  Bc
 x  Ac  Bc
(ii) Similarly we show that Ac  Bc  (A  B)c
18

Same proof with Set builder Notation
(A ∩ B)c = Ac ∪ Bc
• (A ∩ B)c = {x | x  A ∩ B} Definition of complement
= {x |￢(x ∈ (A ∩ B))} Definition of does not belong symbol
= {x |￢(x ∈ A ∧ x ∈ B)} Definition of intersection
= {x | ￢(x ∈ A)∨￢(x ∈ B)} De Morgan’s law
= {x | x  A ∨ x  B} Definition of does not belong symbol
= {x | x ∈ Ac ∨ x ∈ Bc} Definition of complement
= {x | x ∈ Ac ∪ Bc} Definition of union
= Ac ∪ Bc meaning of set builder notation
19

8-Sets-2.ppt

More Related Content

Similar to 8-Sets-2.ppt (20)

More from jaffarbikat (10)

Recently uploaded (20)

8-Sets-2.ppt