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MATH401Chapter4

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CHAPTER 4
Section 4.1 Sets
A set is a collection of objects. There are three ways to denote
a set:
1. List the elements of the set
2. Use recursion (a rule) to generate elements
3. Describe a property of the elements
EXAMPLE
We can describe the same set in three ways:
NOTATION AND TERMINOLOGY
Use ε to symbolize membership: 2 ε S, 128 ε S, 7 ε S
A set that is empty is called the null set or empty set denoted
 or { }. Note that  and {} are not the same.
Standard sets:
Z
Z+
N
Q
R
C
integers
positive integers
integers ≥ 0
rational numbers
real numbers
complex numbers
2
Set A is a subset of B if every element of A is also in B. We write
A ⊂ B or A ⊆ B. The first symbol denotes that A is a proper
subset of B (in other words A ≠ B). The second one includes the
possibility that A = B.
Two sets are equal if they contain the same elements
EXAMPLE
A = {x | x ε N and x ≥ 5}
B = {10, 12, 16, 20}
C = {x |(y)(y ε N and x = 2y)}
Decide whether the following are true or false
B⊆C
B⊄A
A⊂C
26 ε C
{11, 12, 13} ⊆ A
{11, 12, 13} ⊂ C
{12} ε B
{12} ⊆ B
{x | x ε N and x < 20} ⊈ B
5⊆A
⊆B
εA
3
Power set of S is
S, the universal set
SET OPERATIONS
We can carry out operations on sets in ℘(S). The most common
are:
Intersection: A  B =
Union: A  B =
Set Difference: A – B =
Complement: A' =
4
EXAMPLE
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 5, 10}
B = {2, 4, 7, 8, 9}
C = {5, 8, 10}
List elements of:
AC
BC
A'
A–C
B'  (A  C)
{5, 10}  B
5
VENN DIAGRAM
Used to visualize sets and members
A, B are sets
S is universal set
A  B (intersection)
A  B (union)
A' (complement)
A – B (set difference)
C ⊆ B (subset)
EXAMPLE
S = (1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 3, 7, 8, 9}
B = {3, 6, 8}
C = {1, 3, 4, 5}
Fill in all the elements of S in the correct region in the Venn
Diagram below:
AB
B'
A  B'  C
AC
6
CARTESIAN PRODUCT
If A, B are subsets of S then the Cartesian product (or cross
product), A x B is
It is the set of ordered pairs whose first component is from A,
second component is from B. Note that an ordered pair is not a
set.
EXAMPLES
A = {1, 2}, B = {2, 3, 4}
AxB
BxA
A x A = A2
Cartesian products may involve infinite sets. Z x Z = Z2 is the
set of all ordered pairs of integers and R2 is the set of all ordered
pairs of all real numbers which can be represented in the
Cartesian co-ordinate plane.
7
Section 4.2 Counting
Combinatorics deals with counting the number of possibilities in a
certain situation.
SIMPLE EXAMPLES
1. P. 252 Ex. 24
2. An ID code is made up of one letter followed by 2 digits. How
many different codes are there?
3. An ID code is made up one letter followed by 2 numbers. How
many different codes are there if no repetition of digits is
allowed?
MULTIPLICATION PRINCIPLE
If there are n1 possible outcomes for a first event and n2 possible
outcomes for a second event, there are n1n2 possible outcomes
for the sequence of events (both events occur).
The situation is analogous to an intersection of two events.
8
EXAMPLE
How many ways can a President, Secretary and Treasurer be
chosen from a club of 25 people?
ADDITION PRINCIPLE
Suppose two events are disjoint and there are n1 possible
outcomes for the first event and n2 possible outcomes for the
second event. Then the total number of outcomes for one or
other (or both) of the events to occur is n1 + n2
The situation is analogous to a union of two events
EXAMPLE
1. A customer wants to buy a vehicle from a dealer who has 23
automobiles and 14 trucks in stock. How many selections does
the customer have?
2. A customer wants to buy an automobile and a truck from a
dealer who has 23 automobiles and 14 trucks in stock. How
many selections does the customer have?
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Section 4.3
PRINCIPLE OF INCLUSION AND EXCLUSION
|A| = # of elements in A
|B| = # of elements in B
A  B is part of A and B
Suppose we wanted to know the number of elements in A  B.
Maybe |A  B| = |A| + |B|?
This will not work because any elements in A  B will be counted
twice. We need to subtract them out:
This is the Principle of Inclusion and Exclusion for two sets.
EXAMPLE
A pollster queries 35 voters all of whom support referendum 1 or
referendum 2 or both. 14 support referendum 1, 26 support
referendum 2. How many voters support both?
A = {voters for referendum 1}
B = {voters for referendum 2}
|A| = 14, |B| = 26
|A  B| = 35, |A  B| = ?
10
PRINCIPLE OF INCLUSION AND EXCLUSION FOR 3 SETS
EXAMPLE
P. 266, Ex 41
A = {students who eat sausage}
B = {students who eat pepperoni}
C = {students who eat extra cheese}
Use |A  B  C| = |A| + |B| + |C| - |A  B| - |A  C| - |B  C| +
|A  B  C| and substitute the values from the question:
|A  B  C| = 13 + 10 + 12 – 4 – 5 – 7 + 3
= 22
11
PIGEONHOLE PRINCIPLE
If more than k items are placed into k bins then at least one bin
contains more than one item
EXAMPLES
1. How many people must be in a room to ensure that 2 people
have first names that begin with the same letter?
2. How many times must a die be rolled to guarantee the same
value twice?
3. How many times must be a die rolled to guarantee the same
value three times?
12
Section 4.4: Permutations and Combinations
PERMUTATIONS
A permutation is an arrangement or ordering of objects
EXAMPLES:
1. How many ways to arrange 4 books on a shelf?
2. There are 8 men in a race. How many ways can gold, silver,
bronze be awarded?
GENERAL PERMUTATION PROBLEM
How many ways to arrange r objects from a group of n objects?
NOTATION
We denote the general situation by P(n, r) or nPr or Prn
P(n, 0) =
P(n, 1) =
P(n, n) =
EXAMPLES
1. There are 4 books on a shelf and all 4 are being arranged.
So n = 4, r = 4.
13
2. There are 8 men and 3 are being arranged. So n = 8, r = 3
3. How many ways can 6 people be seated in a row of 6 chairs?
4. How many 3 letter strings can be formed from the word
"winter" with no repetition of letters in a string?
COMBINATIONS
A combination occurs when a group of objects is selected with no
consideration for the order of the objects.
EXAMPLES
1. How many ways to select 4 books from a shelf of 16 books?
2. How many ways to select 5 cards from a full deck?
GENERAL COMBINATION PROBLEM
How many ways to select r objects from a group of n objects?
We denote the general situation by C(n, r) or nCr or Crn
C(n, 0) =
C(n, 1) =
C(n, n) =
14
EXAMPLES
1. n = 16, r = 4
C(16, 4) =
2. n = 52, r = 5C(52, 5) =
3. 8 athletes compete in an event where three will be declared
winners. In how many ways can the winners be chosen?
15
Section 4.5 Binomial Theorem
PASCAL'S TRIANGLE
1
1
1
1
2
3
1
1
1
4
5
1
3
6
10
1
4
10
1
5
1
Every entry is the sum of the two (or one) numbers directly above
it. If row 0 is the first row then we can prove than row n looks like:
C(n, 0)
C(n, 1)
C(n, 2)
C(n, 3) …………… C(n, n – 1)
C(n n)
Each row is symmetric.
BINOMIAL THEOREM
(a + b) = C(n, 0)anb0 + C(n, 1)an – 1b1 + C(n, 2)an – 2b2 +…..………
+ C(n, n – 1)a1bn – 1 + C(n n)a0bn
𝑛
=∑𝑘=0 𝐶(𝑛, 𝑘)𝑎𝑛−𝑘 𝑏𝑘
16
EXAMPLES
(x + 2)3
(2x + 3y)4
(2 – x)5
(4y – 3)3
In the expansion of (5x + 7y)37 write down the term involving
x20y17
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