Download Day 01 - Introduction to Sets

IB Math Studies
Topic 3 Day 1 – Introduction to Sets
Name ________________________
Date ________________ Block ____
A set is ____________________________________________________________________________.
 SET NOTATION
 Sets are denoted with a capital letter. The items, names, objects or numbers are enclosed by { } .


is the symbol denoting ‘is an element of’
ex.
 is the symbol denoting “is not an element of’
A = {all even numbers},
therefore
3
A
 The “bar” symbol “|” is read “such that”
ex.
F = {x| x is odd} would be read “F is the set of all x, such that x is odd.”
 The empty set { }, also called the null set, is a set with no elements and is written as


SETS OF NUMBERS
n = {0, 1, 2, 3, 4, ….} CARDNIAL (NATURAL) NUMBERS
Z = {…-3, -2, -1, 0, 1, 2, 3, ….} INTEGERS
Q = {m | m =
where a and b
Z and b
0} RATIONAL NUMBERS
R = {all numbers on the real number line} REAL NUMBERS
The ________________ ________________ of a set A, n(A), represents the number of
________________ in set A.
A finite set has a __________________ number of elements and an infinite set has an infinite number of
elements.
The _____________ _________ is denoted as { } or  .

SUMMARY OF TYPES OF SETS
TYPE OF SET
Null Set
(Empty Set)
Finite Set
Infinite Set
Definition
A set that has no elements
It is denoted by { } or 
A set whose elements can be
counted.
A set for which it is impossible to
list the elements.
Example
{x|x2 + 1 = 0, x  R }
The set {1, 2, 4, 5, 10, 22}
contains 6 elements
The set {y | y is a real number}
cannot be counted, as we cannot
write a list of all its elements.
THE ALGEBRA OF SETS
 Set A is said to be equal to set B if both sets are identical, that is, they contain the same elements. We write
this as A = B.
 Set A is said to be equivalent to set B if n(A) = n(B). We write this as A  B.
ex. Set A = {10, 20, 30, 40, 50} and set B = {a, b, c, d, e}. Set A is equivalent to set B. They are said to have a
one-to-one correspondence. (*NOTE: Equivalent means that they have the same number of elements).
10
a
20
b
30
c
40
d
50
e
 Subsets: Set B is a subset of set A if all elements in set B are contained in set A. Every set is a subset of itself.
ex. the set of integers is a subset of the set of real numbers because every integer is also a real number
 denotes ‘is a subset of’
 denotes ‘is not a subset of’
 Proper Subsets: Set B is a proper subset of set A if all the elements of B are contained in set A, but the two
sets are not the same. A set IS NOT a proper subset of itself.
 denotes ‘is a proper subset of’
 denotes ‘is not a proper subset of’
ex.
Let A = {green, yellow, blue}
The following are subsets of A:
{ }, {green}, {yellow}, {blue}, {green, yellow}, {green, blue}, {yellow, blue}, {green, yellow, blue} There
are 8 subsets (including the empty set)
The following are proper subsets of A:
{ }, {green}, {yellow}, {blue}, {green, yellow}, {green, blue}, {yellow, blue}
There are 7 proper subsets (the set A itself cannot be a proper subset)
CORRECT SYMBOL USAGE IN SETS
Symbol


Correct Usage
2  {1, 2, 4}
{2}  {1, 2, 4}
Incorrect Usage
{2}  {1, 2, 4}
2  {1, 2, 4}
 Number of Subsets and Proper Subsets:
If n(A) = p, then there are 2P subsets of A and 2P - 1 proper subsets of A.
 Intersection: The intersection of two sets is the set of elements that they have in ____________ and is
denoted by the symbol  .
ex.
A = {even numbers}
B = {prime numbers}
A  B = {2}
 Union: The union of two sets is the combination of all elements in ____________ sets (each element only being
listed once) and is denoted by the symbol  .
ex.
A = {the letters in the word ‘mathematics’}
B = {a, b, c, d, e}
A  B = {a, b, c, d, e, h, i, m, s, t}
Number of elements in a union of two sets:
n(A  B) = n(A) + n(B) - n(A  B)
**THIS IS AN IMPORTANT CONCEPT IN SET THEORY! **
 Complements of Sets and the Universal Set: The universal set is denoted by U. The complement of set A is all
of the elements that are in the universal set but are not also in set A. The complement of A is denoted by A’
ex.
U = {the letters in the word ‘universal’}
A = {a, e, i}
A’ = {l, n, r, s, u, v}
A  A’ = U
A  A’ = 
the union of a set and its complement is the universal set
the intersection of a set and its complement is the empty set.
n(A) + n(A’) = n(U)

The sum of the number of elements in a set and its complement equals the
total number of elements in the universal set.
VENN DIAGRAMS
Disjoint sets – set
are disjoint if they
have no elements in
common
Intersecting Sets –
sets with elements in
common
U
Difference – A\B
the set of all element
in A and not in B
U
B
A
Subsets – a set
within a set
A
U
B
B
A  B = 
A  B
A
U
A
B
A  B = B
A\B
Examples: Shade the area of the Venn diagram noted by the set notation below the universal set.
1.
INTERSECTION
Disjoint sets
Intersecting sets
U
U
A
B
A
A  B
2.
Subsets
B
U
A
B
A  B
A  B
UNION
Disjoint sets
Intersecting sets
B
A
A  B
Subsets
U
U
A
B
A  B
U
A
B
A B
3.
COMPLEMENT
Disjoint sets
Intersecting sets
Subsets
U
U
B
A
A
A’
4.
B
A’
COMPLEMENTS and INTERSECTION
Intersecting sets
Subsets
U
U
B
A
A
A’  B’
ex
A
B
A’
Disjoint sets

U
U
B
A
B
A’  B
A  B’
APPLICATIONS
A.
The classes offered at a summer camp are Crafts(C), Volleyball (V) and Swimming (S). The Venn
diagram shows the numbers of children involved in each activity.
U
C
V
8
1
5
3
2
S
(a)
(b)
(c)
(d)
4
6
4
How many children are in only the Crafts class?
How many children are in both the Crafts and Volleyball class?
How many children are not in the Crafts class?
Shade the part of the Venn diagram that represents the set C’  S.
____________
__________
__________
What does this represent?
ex
B:
During a two week period, Murielle took her umbrella with her on 8 days, it rained on 9 days, and
Murielle took her umbrella on five of the days when it rained.
(a)
Display this information on a Venn diagram:
(b)
Find the number of days that Murielle did not take her umbrella and it rained.
(c)
Find the number of days that Murielle did not take her umbrella and it did not rain.