Review of Sets and Set Properties
Consider Each of the following Situations
1. A company employs 20 women, 4 accountants, and 6 sales agents.
How many individuals are employed by the company ? _____________
How many of the accountants and sales agents are women ? _______
If a group of three is selected , what is the probability that of the three selected 2 are women ? __________
2. An investment analyst describes 10 possible companies to invest in. Last year he was right on 90 out of 100
companies. You pick one at random. Invest in it. What is the likelihood that you picked a “good” investment ?
3. A new pill for diabetics claims to lower sugar levels to a normal level in all but 8 % of all cases. A group of 500
takes the pill to verify their claims. How many individuals should be expected to have a lower sugar level ( normal ) ?
4. Grades are known to be normally distributed with mean 70 and variance 9. A student is selected at random
What is the probability that the student has a grade of 80 or higher ?
5. The cost in cleaning up a river is given by g(x) = , where x represents the percent of purity.
What will be cost to return the river to 100 % clean ?
2. Sets
A set is a collection of numbers or objects that have a welldefined property in common.
Which of these represent welldefined sets ?
ex. Set of all smart students in class  ____________
ex. Set of all students with long hair  ___________
ex. Set of all good fruit  _______________
ex. The set of all students enrolled in math 1312.050 on Jan. 25 during the spring semester of 2005.
There are times that we can list all members of a set  { .... }
{ a, e, i, o, u } { A, B, C, D, F , W} { 2, 4, 6, 8 }
But other times we can not:
although by listing several members – we get the idea of what the set contains
{ 1, 3, 5, 7, ... }, { red, blue, gray, purple, green, violet,... } { Joe, Jim, John, Jeff, Jack,... }
We can use what’s is called setbuilder notation.
This way we do not write every member of a set – just indicate what the members look like.
{ x  x represents a student enrolled in math 1312.050 – spring 2005 on Jan. 25}
{ x  x represents a brand of car that you have driven }
{ x x represents the amount of money in a students bank account }
In some cases – one form may be better than the other.
{ a, e, i, o, u } { x  x represents a student currently enrolled at A.S.U }
The members of the set are called elements of the set. We use the symbol to express the fact that an object is a
member of a set.
capital letters: sets lower case letters: elements
A = { a, b, c } or B = { 5, 10, 15 ..... } , C = { x  x is a positive real number }
We say a A, 5 B, d A, or 7 B
Is 0 C ? ___________ Is there a smallest member of C ? _____________
Sets of numbers:
N = set of natural numbers = set of counting numbers = set of positive integers
= { x  x set of natural numbers } = { 1, 2, 3, 4, .... }
W = set of whole numbers = set of nonnegative integers
= { x  x set of whole number } = { 0, 1, 2, 3, ... }
I = set of integers =
= { x  x set of integers } = { ... 3, 2, 1, 0, 1, 2, 3, ... }
Q = set of rational numbers
= { x  x set of rational numbers } = { m/n  m and n are integers with n ≠ 0 }
We can not list the members of this set as nicely as we did the first three. The best we can do is
say the above statements and provide examples.
 3 = 3/1, 0 = 0/6, 4 = 4/1, 2/3, 5/17, 0.24, 0.11111... ( 1/9 ) , ...
Q ^{/} = set of irrational numbers → examples:
π, e, , 0.1010010001... , ...
R = set of real numbers : consists of the set of rational and irrational number – no other number
Sets
Def. A set is a collection of objects having a well defined property(characteristic) in common.
 which of these can be classified as sets ?
_________ Set of all female students in this class .
____________ the set of all companies with smart chairman
___________ set of all companies with executives who were given a salary exceeding 10million dollars / year
At times it is useful to list all members of a set but when this is not feasible or desired we can also write sets in
setbuilder notation
4. A = { x : x is a whole number less than four } = _____________________________
5. B = { x : x is an integer with  x  < 2 } = ____________________________
6. C = { x : x is a current Fortune 500 company } = ______________________________________
7. D = { x : x represents the name of a company that has not had a posted a losing quarter in the last 20 years }
= _______________________________
Notice this last example. If no such company exists, then this is a set that is called an __________, or the ________
Def.1.1
A ___________ is a collection of objects having a well defined property in common.
Def. 1.2
The set that contains all objects under consideration is called the ______________________
Depending on the Experiment – the universal set will vary.
ex. Find all x’s so that
A = { x  x^{2} = 4 } B = { x  x < 0 } C = prime numbers
Def.1.3
The objects – or members  of a set A are called _________________ of the set.
ex. If A = { 1, 2, 3 } we write 2 A and say “ ____________________________________________”
we write 5 A and say “ _____________________________________________”
Venn Diagrams are used to illustrate sets and their relationship to each other and the universal set U. Consist of rectangles
and “circles” to represent the sets and the universal set.
The first relationship between sets:
Def. 1.4
Let A and B be any two sets. We say A is a ___________________ of B provided every element of A is also in B.
Note: If A is not a subset of B ( A B ) , then there exists an element in A that is not in B.
Example.
A = {x : is a natural number < 5 },B = { x : x is an even prime number }, C = {x: x is the largest negative whole number }
List the members of each set.
A = _______________________________ B = ___________________________
C = ____________________
Are any of the sets above subsets of another set listed ? ______________________
Example.
A = { set of students in class that are male }, B = { set of students in class that are over 20 years of age }
C = { set of students in class that have a daytime job }
In general, which of these sets can be classified as subsets of each other ?
What is the universal set ?
Def.1.5
Let A and B be any two sets of some universal set U. We say that A and B are said to be ____________ provided
A B and B A.
ex. Is { a, a, b } { a, b } ? ____________ What about { a, b } { b, a } ? ________
Conclude that 1. Repetitions do not count – if there are repetitions, we can rewrite the set with no repetitions
2. Order does not matter – order that the elements are written is not important
We want to be able to count the numbers of subsets of a given set.
ex. How many subsets does the set
{ a } have ? _________ What about the set {1, 2 } ? ____________
Properties of sets
1. We use to compare an element to a set
2 A is acceptable or 5 B is also acceptable but { 1, 2 } A is not. Why ? ____________________
examples:
a) A = { x  x is an so that x^{2} > 0 }
Is  2 A ? _____________ Is 25 A ? ____________
Is there any integer that is not an element of A ?
b) B = set of all students in class that are enrolled in math 1312 but have not taken math 1311.
Is B empty ? ___________
2. We use to compare a set to a set
{ 1, 2 } A or { 1} { 1, 2 } or the set of natural numbers is a subset of the set of integers , N I
but we can not use 2 A. Why ? _________________________________________
ex. List some of the subsets of the set A = set of all letters of the alphabet that are considered vowels.
ex. True or False. 0 set of whole numbers . ___________________
3. The universal set U contains all objects under consideration and the empty set (the null set ) contains no object at all
U  the universal set ,
or { }  the null set, the empty set → we do not use { } to represent the empty set
example: let A = { x  x ^{2} = 4 }
example: let the set B represent the students that voted in the last local elections.
4. A for any set A , the null set is a subset of any set including itself ( )
for any set A, A A.
Def.1.6
Let A be a subset of B. If B is not a subset of A, then A is not equal to B and A is said to be a proper subset of B.
ex. Consider the set of all individuals in this classroom ( set A ) and the set of students in this classroom (B ) . How are
these two sets related ?
Are they subsets of each other or are they proper subsets ?
ex. Consider the set of all male students in class today ( M ) and the set of all students in class still awake and listening ( A ).
How are these two sets related ?
ex. Let U = set of all integers less than 10 , A = { x  x^{2} = 16 } , and B = { x  = x }
Which of these statements is true ? A B B A B = A
Are either A or B proper subsets of each other ?
Counting subsets:
How many subsets does the set B = { even prime numbers }
How many subsets does the set B = { 1, 2 } have ? ___________
How many subsets does the set A = { a, b, c } have ? _________
How many subsets does the set A = { a, b, c, d have ? _________
In general how many subsets does a set with n objects have ? _____________ how many proper subsets ? ________
A committee is to be appointed from a group of five.
The committee can consist of 1 , 2, 3, 4, or 5 individuals. How many distinct committees are possible ?
Def.1.7 (Disjoint)
Let A and B be any two nonempty sets. If A and B have no element in common ,
then A and B are said to be ______________ .
Use Venn Diagrams to illustrate sets that are
disjoint sets sets that are not disjoint subsets of each other
ex. Which of the following pairs of sets are disjoint ?
__________________ 1) A = { x : x is an even natural number } B = { x : x is an odd whole number }
__________________ 2) C = set of all nonnegative integers D = set of all nonpositive integers
___________________ 3) set of all
ex Let C = { x : x is a rational number }, D = { x : x is an integer } E = { x : x is a whole number }
F = { x : x a nonnegative integer } G = { x : x is a natural number }
Are any of these sets equal ?
Which is the “largest” set ? Is there one ? ( largest in the sense that it contains all of the other sets )
Def.1.8 ( Complement )
Let A be any set of some universal set U. We define the complement of A, A^{/} as the set that contains
all objects in U that are not in A.
ex.
ex. Let U = { x : x is a whole number } with A = { x : x is positive } , B = { x : x is even whole number }
Find A^{/} = ____________________________ B^{/} = ______________________________
Let A = set of all days in which rain of 1 inch or less fell in San Angelo.
A ^{/} =
Let B = set of all students in class with at least one ring.
B ^{/} =
Let C = set of all students that are at least 40 years of age or older
C ^{/} = ....
Let D = set of all four card hands with at least one diamond.
D^{/} =
Def. 1.9( and )
Let A and B be any two sets of some universal set U. We define
1) the intersection of A and B (written A B ) as
A B = set of all objects (elements) that are in A and at the same time they are also in B
 each element in A∩B must be classified as being part of A and at the same time part of B
ex. a couple’s mutual friends ex.
2) the union of A and B ( written A B) as
A B = set of all elements that are in A, in B, or in both A and B (either A or B )
( they are in at least one of the two sets but not necessarily in both sets – although they can be)
ex. when a couple gets married and they bring in all their property.
ex. Let U = { all positive integers less than 5 }, A = { x : x^{2} = 4 }, B = { x : x < 3 }, C = { x : x > 2 }
1) union: What is A C = ___________ B C = ______________ A B C = ___________
2) intersection: A B = ___________, A C = ____________ , B ∩ C
More properties of sets.
a. complements
1) (A^{/} ) ^{/} = _________________ 2) ^{/} = _____________ 3) U ^{/} = ___________
b.
1) ( A A^{/} ) = _________ 2) ( A A^{/} ) = ________
c. De Morgan’s Law:
1) ( A B ) ^{/} = A^{/} B^{/} 2) ( A B ) ^{/} = A^{/} B^{/}
d. Commutative, associative, and distributive
1) A u B = B u A and A n B = B n A
2) ( A u B ) u C = A u ( B u C ) and ( A n B ) n C = A n ( B n C )
3) A n ( B u C ) = ( A n B ) u ( A n C ) and .......
Any of these laws can be proved by using Venndiagrams.
Review of Sets:
1. The set that contains all elements under consideration is called the ________________________ set.
2. The sets A and B are said to be ________________________ provided A ∩ B =
3. If x A and x B, then x _________ and x ___________
( either A or B) ( both A and B )
4. The ____________ of sets A and B consists of all objects that are in A or in B or in both A and B.
5. If A = { 1, 2, 3 } and B = { 2, 3, 4 }, then another name for the set { 2, 3 } would be ____________
6. If the universal set U = { 1, 2, 3, ..., 10 } and A = { x  x^{2} ≥ 4 }, then the complement of A = A ^{/} = ______________
7. Find the union of A = {a, e, i , o, u } and B = { b, c, d, f, g, h, .... } in set builder notation.
8. Draw a Venn Diagram for each of the following sets. Make sure to include the universal set.
a) A B^{/} = b) A ∩ ( B ∩ C ^{/} )
9. Prove or Disprove by shading the sets that correspond to each side and indicating whether they are equal or not.
A B ^{/} = ( A^{/} ∩ B ^{/} ) A . Provide your answer (work) on the back of this page.
10. True or False.
______________ a)  2 { x  x^{2} = 4 } ______________ b) 0 { x  x is a whole number }
______________ c) { 1, 2 } ______________ d ) A, for any set A
_______________e) ( A B ) ^{/} = A ^{/} ∩ B ^{/} ______________ f) A A, for any set A
_______________ g) U ^{/} = ______________ h ) A A^{/} =
11. How many subsets does the set { a, e, i, o, u } have ? ____________ How many proper subsets ? __________
12. Two sets A and B are said to be equal provided 1) _____________________ and 2) ________________
13. Find the absolute value of each of the following
a) ___________________ = b) = ______________ c) =
14. A = { x  x^{2} = 4 } and B = { x  1 < x < 3 }
a) Can A be a subset of B ? Why or Why not ?
b) Can B be a subset of A ? Why or Why not ?
c) Could they be equal ? Why or Why not ?
