1. Partially borrowed from Florida State University
Set and Set Operations
CMPS Chapter 2
Partially borrowed from Florida State University
Some developed by Dr. H

2. Introduction A set is a collection of objects
The order of the elements does not matter
Objects in a set are called elements
A well – defined set is a set in which we can determine if an element belongs to that set.
Examples:
The set of all movies in which John Wayne appears is well – defined.
The set of all courses taught by Dr. Halverson
The set of best TV shows of all time is not well – defined. (It is a matter of opinion.)

3. Notation Usually denote a set with a capital letter.
Roster notation is the method of describing a set by listing each element of the set.
Example: Let C = set of all even integers less than 12 but greater than zero.
C = {2, 4, 6, 8, 10} = {10, 6, 8, 4, 2}
Example: Let T = set of all courses taught by Dr. Halverson in summer 2014.
T = {CMPS 1013}

4. More on Notation Sometimes impossible list all elements of a set.
Z = The set of integers . The dots mean continue on in this pattern forever and ever.
Z = { …-3, -2, -1, 0, 1, 2, 3, …}
Dots can ONLY be used if the pattern is unmistakable
W = {0, 1, 2, 3, …} = the set of whole numbers.

5. Set – Builder Notation
Set-Builder Notation: specify the rule that determines set membership
First: indicate type of elements in set
Second: specify distinguishing rule
V = { people | citizens registered to vote in Wichita County}
A = {x is a real number | x > 5}
The symbol | is read as “such that”

6. Special Sets of Numbers
N = The set of natural numbers.
= {1, 2, 3, …}.
W = The set of whole numbers.
={0, 1, 2, 3, …}
Z = The set of integers.
= { …, -3, -2, -1, 0, 1, 2, 3, …}
Q = The set of rational numbers.
={x| x=p/q, where p and q are elements of Z and
q ≠ 0 }
H = The set of irrational numbers.
R = The set of real numbers.
C = The set of complex numbers.

7. Universal Set and Subsets
Universal Set (denoted by U) - set of all possible elements used in a problem
Universal set  Whole numbers (if counting)
Universal set  Rational numbers (if measuring)
Subset: B A if every element of B is also an element A
Example A={1, 2, 3, 4, 5} and B={2, 3}
Let S={1,2,3}, list all the subsets of S.
The subsets of S are , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

8. The Empty Set Empty Set: set containing no elements; zero elements
denoted as { } or
Empty Set  of all sets
Do not be confused by this question:
Is this set {0} empty?
It is not empty! It contains one element - zero

9. Intersection of sets Intersection of sets A & B is denoted A ∩ B
A ∩ B = {x| x is an element of A and x is an element of B}
A={1, 3, 5, 7, 9} & B={1, 2, 3, 4, 5}
A ∩ B = {1, 3, 5}
B ∩ A = {1, 3, 5}

10. Union of sets
Union of two sets A, B is denoted A U B and is defined A U B = {x| x is in A or x is in B}
A={1, 3, 5, 7, 9} & B={1, 2, 3, 4, 5}
A U B = {1, 2, 3, 4, 5, 7, 9}.
NOTE:
Order of listing sets does not matter
Never repeat elements

11. Difference of Sets Difference of sets A & B denoted A-B
A-B = {x|x is in A but x is NOT in B}
Like subtraction
A={1, 3, 5, 7, 9} & B={1, 2, 3, 4, 5}
A-B = {7, 9}
B-A = {2, 4}

12. Mutually Exclusive Sets
Two sets A and B are Disjoint (aka Mutually Exclusive) if A ∩ B =
I.E. No elements in common
Think of this as two events that can not happen at the same time.

13. Complement of a Set Complement of set A is denoted by Ᾱ or by Ac
Ᾱ = {x | x is in the universal set but x is not in set A}
U={1,2,3,4,5} & A={1,2},
Ᾱ = {3,4,5}

14. Cardinal Number |{ } | = ??? (Cardinality of the empty set?)
Cardinality of a set is the number of elements in the set and is denoted by |A| or n(A)
A={2,4,6,8,10}, then |A|=5.
Cardinality formula
|A U B|=|A| + |B| – |A∩B|
|{ } | = ??? (Cardinality of the empty set?)

15. Theorem 2.1 (pg. 43) Commutative Law Associative Law Distributive Law
A U B = B U A and B ∩ A = A ∩ B
Associative Law
(A U B) U C = A U (B U C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive Law
A U (B ∩ C) = (A U B) ∩ (A U C)
A U (B ∩ C) = (A ∩ B) U (A ∩ C)
See others on Page 43

16. Ordered Pair Order is significant (a,b) ≠ (b,a)
An Ordered Pair of elements a & b is denoted (a,b)
Order is significant
(a,b) ≠ (b,a)

17. Cartesian Product Cartesian Product of sets A & B Denoted A X B
is the set consisting of all ordered pairs (a,b) where a is an element of A and b is an element of B
A X B = { (a,b)| a is in A & b is in B}
If |A| = 3 and |B| = 7, what is |A X B|?
Can you list them?

18. De Morgan’s Laws (pg. 45) See Page 45 – Memorize Study Proofs also!
And YOU prove the part not proven in the book

19. Homework – 2.1
Pages 46 & 47
1-8, 13-28