Mathematics Sets and Logic Week 1

1. Chapter 1 :
Introduction to Set and Logic
SM0013 Mathematics I
Khadizah Ghazali
Lecture 1 – 25/05/2011

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LEARNING OUTCOMES
At the end of this chapter, you should be able
to :
know what set & logic are about
define some basic terminologies in set &
logic
identify relations between pairs of sets
use Venn diagrams & the counting
formula to solve set equations
build & use the truth table

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Outline (A)
1.1 Symbols in set & logic
1.2 Set Terminology and Notation
1.3 Operations on set
1.4 Algebra of sets
1.5 Finite Sets and Counting
Principle.

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Outline (B)
1.6 Statements and Argument
1.7 Combining, Conditional and
Biconditional Statements
1.8 Truth Table
1.9 Logic of Equivalent

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SECTION 1.1
Symbols in set & logic
Symbols is the central part in set & logic. Here some
important symbols :

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∋∋

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Here is a list of sets that we will refer
to often, and the symbols are standard.

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Example 1.1Example 1.1

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Example 1.2Example 1.2

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Example 1.3Example 1.3

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SECTION 1.2
Set Terminology & Notation
George Cantor (1845-1915), in 1895,
was the first to define a set formally.
Definition : Set
A set is a group of things of the same kind
that belong together.
The objects that make up a set are
called
elements or members of the set.

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Some Properties of Sets
The order in which the elements are
presented in a set is not important.
A = {a, e, i, o, u} and
B = {e, o, u, a, i} both define the same set.
The members of a set can be anything.
In a set the same member does not appear
more than once.
F = {a, e, i, o, a, u} is incorrect since the element
‘a’ repeats.

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Some Notation
Consider the set A = {a, e, i, o, u} then
We write “‘a’ is a member of ‘A’” as:
a ∈ A
We write “‘b’ is not a member of ‘A’” as:
b ∉ A
Note: b ∉ A ≡ ¬ (b ∈ A)

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Set representation:
There are 4 ways to represent a set.
1. Set may be represented by words, for
example:
A = the first three natural numbers greater
than zero
B = the colors red, white, blue, and green

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Set representation (cont.):
2. Another way to represented a set is to list
its elements between curly brackets (by
enumeration).
~ is called the roster method, for
example:
C = {1, 2, 3}
D = {red, white, blue, green}

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Set representation (cont.):
3. Another kind of set notation commonly used is
set-builder notation, for example:
The set E of a natural number less than 4 is
written as

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Set representation (cont.):
4. A set can also be represented by a Vann
diagram
A pictorial way of representing sets.
The universal set is represented by the
interior of a rectangle and the other sets
are represented by disks lying within the
rectangle.
E.g. A = {a, e, i, o, u} a
e
i
o
u
A

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Equality of two Sets
A set ‘A’ is equal to a set ‘B’ if and only if both
sets have the same elements. If sets ‘A’ and ‘B’
are equal we write: A = B. If sets ‘A’ and ‘B’ are
not equal we write A ≠ B.
In other words we can say:
A = B ⇔ (∀x, x∈A ⇔ x∈B)
E.g.
A = {1, 2, 3, 4, 5}, B = {2, 4, 1, 3, 5}, C = {1, 3, 5, 4}
D = {x : x ∈ Z ∧ 0 < x < 6}, E = {1, 10/5, , 22, 5}
then A = B = D = E and A ≠ C.
9

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Universal Set and Empty Set
The members of all the investigated sets in
a particular problem usually belongs to
some fixed large set. That set is called the
universal set and is usually denoted by ‘U’.
The set that has no elements is called the
empty set and is denoted by Φ or {}.
E.g. {x | x2 = 4 and x is an odd integer} = Φ

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Cardinality of a Set
The number of elements in a set is
called the cardinality of a set. Let ‘A’ be
any set then its cardinality is denoted
by |A| @ n(A)
E.g. A = {a, e, i, o, u} then |A| = 5.

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Subsets
Set ‘A’ is called a subset of set ‘B’ if and
only if every element of set ‘A’ is also
an element of set ‘B’. We also say that
‘A’ is contained in ‘B’ or that ‘B’ contains
‘A’. It is denoted by A ⊆ B or B ⊇ A.
In other words we can say:
(A ⊆ B) ⇔ (∀x, x ∈ A ⇒ x ∈ B)

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Subset cont.
If ‘A’ is not a subset of ‘B’ then it is
denoted by A ⊆ B or B ⊇ A
E.g. A = {1, 2, 3, 4, 5} and B = {1, 3} and
C = {2, 4, 6} then B ⊆ A and C ⊆ A
1 3
5
2
4
6
B
A
C

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Some Properties Regarding
Subsets
For any set ‘A’, Φ ⊆ A ⊆ U
For any set ‘A’, A ⊆ A
A ⊆ B ∧ B ⊆ C ⇒ A ⊆ C
A = B ⇔ A ⊆ B ∧ B ⊆ A

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Proper Subsets
Notice that when we say A ⊆ B then it
is even possible to be A = B.
We say that set ‘A’ is a proper subset of
set ‘B’ if and only if A ⊆ B and A ≠ B.
We denote it by A ⊂ B or B ⊃ A.
In other words we can say:
(A ⊂ B) ⇔ (∀x, x∈A ⇒ x∈B ∧ A≠B)

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Venn Diagram for a Proper
Subset
Note that if A ⊂ B then the Venn diagram
depicting those sets is as follows:
If A ⊆ B then the disc showing ‘B’ may
overlap with the disc showing ‘A’ where in
this case it is actually A = B
B A

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Power Set
The set of all subsets of a set ‘S’ is called the
power set of ‘S’. It is denoted by P(S) or 2S.
In other words we can say:
P(S) = {x : x ⊆ S}
E.g. A = {1, 2, 3} then
P(A) = {Φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
Note that |P(S)| = 2|S|.
E.g. |P(A)| = 2|A| = 23 = 8.

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???
Is the empty set, Φ a subset of {3, 5, 7}?
If we say that Φ is not a subset of {3, 5, 7},
then there must be an element of Φ that does
not belong to {3, 5, 7}.
But that cannot happen because Φ is empty.
So Φ is a subset of {3, 5, 7}.
In fact, by the same reasoning, the empty set
is a subset of every set.

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SECTION 1.3
Operations on Set
Complement :
The (absolute) complement of a set ‘A’ is
the set of elements which belong to the
universal set but which do not belong to A.
This is denoted by Ac or Ā or Á .
In other words we can say:
Ac = {x : x∈U ∧ x∉A}

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Venn Diagram for the
Complement
A
Ac

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Set Complementation
If U is a universal set and A is a subset of U,
then

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∪nion
Union of two sets ‘A’ and ‘B’ is the set of all
elements which belong to either ‘A’ or ‘B’ or
both. This is denoted by A ∪ B.
In other words we can say:
A ∪ B = {x : x∈A ∨ x∈B}
E.g. A = {3, 5, 7}, B = {2, 3, 5}
A ∪ B = {3, 5, 7, 2, 3, 5} = {2, 3, 5, 7}

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Venn Diagram Representation for
Union
B
A
A ∪ B
3 57
2

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I∩tersection
Intersection of two sets ‘A’ and ‘B’ is the set
of all elements which belong to both ‘A’ and
‘B’. This is denoted by A ∩ B.
In other words we can say:
A ∩ B = {x : x∈A ∧ x∈B}
E.g. A = {3, 5, 7}, B = {2, 3, 5}
A ∩ B = {3, 5}

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Venn Diagram Representation for
Intersection
B
A
A ∩ B
3
57
2

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Difference
The difference or the relative complement of a
set ‘B’ with respect to a set ‘A’ is the set of
elements which belong to ‘A’ but which do not
belong to ‘B’. This is denoted by A B.
In other words we can say:
A B = {x : x∈A ∧ x∉B}
E.g. A = {3, 5, 7}, B = {2, 3, 5}
A B = {3, 5, 7}{2, 3, 5} = {7}

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Venn Diagram Representation for
Difference
B
A
A B
3 57
2

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Disjoint Sets
If two sets A and B have no elements in
common, that is, if
then A and B are called disjoint sets.
Notice that the circles corresponding to A and
B not overlap anywhere because A∩B is
empty.
A ∩ B = Φ
B A

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Some Properties
A ⊆ A ∪ B and B ⊆ A ∪ B
A∩B ⊆ A and A∩B ⊆ B
|A ∪ B| = |A| + |B| - |A∩B|
A ⊆ B ⇒ Bc ⊆ Ac
A B = A ∩ Bc
If A ∩ B = Φ then we say ‘A’ and ‘B’ are
disjoint.

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SECTION 1.4
Algebra of Set
In section 1.3 we saw that, given two sets A
and B, the operations union and intersection
could be used to generate two further sets
These two new sets can then be combined
with a third set C, associated with the same
universal set U as the sets A and B, to form
four further sets
∪ ∩A B and A B
( ) ( ) ( ) ( )∪ ∪ ∩ ∪ ∪ ∩ ∩ ∩C A B , C A B , C A B , C A B

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And the compositions of these sets are clearly
indicated by the shaded portion in the
Venn diagrams

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Algebraic Laws on Sets:
Sets operations of union, intersection and
complement satisfy various laws
(identities).
Let U be the universal set and A, B and C
are subsets of U.

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Algebra of Sets
Idempotent laws
A ∪ A = A
A ∩ A = A
Associative laws
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

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Algebra of Sets cont.
Commutative laws
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Distributive laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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Algebra of Sets cont.
Identity laws
A ∪ Φ = A
A ∩ U = A
A ∪ U = U
A ∩ Φ = Φ
Involution laws
(Ac)c = A

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Algebra of Sets cont.
Complement laws
A ∪ Ac = U
A ∩ Ac = Φ
Uc = Φ
Φc = U

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Algebra of Sets cont.
De Morgan’s laws
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
Note: Compare these De Morgan’s laws
with the De Morgan’s laws that you will
find in logic and see the similarity.

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Proofs
Basically there are two approaches in
proving above mentioned laws and any
other set relationship
Mathematical Notation
Using Venn diagrams

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Example 1.4 :Example 1.4 :

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Solution : a)Method 1;

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Solution : a)Method 2;

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Solution :
b)

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SECTION 1.5
Finite Sets & Counting Principle
Definition : Finite set
A set is said to be finite if it contains
exactly p elements. Otherwise a set is
said to be infinite.

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Number of the elements in a set;
is determined by simply counting the
elements in the set.
If A is any set, then n(A) or |A| denotes the
number of elements in A.

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Another result that is easily seen
to be true is the following cases :

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Case III : Inclusion-Exclusion
Principle

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Case IV:

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Example 1.5 :Example 1.5 :

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Example 1.6 :Example 1.6 :

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Example 1.7 :Example 1.7 :

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