Set in discrete mathematics

A set is said to contain its elements.
A set is an unordered collection of objects, called elements or
members of the set.
{1, 2, 3} is the set containing “1” and “2” and “3.” list the members
between braces.
{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant.
{1, 2, 3} = {3, 2, 1} since sets are unordered.
{1,2,3, …, 99} is the set of positive integers less than 100
{1, 2, 3, …} is a way we denote an infinite set (in this case, the
natural numbers).
∅ = {} is the empty set, or the set containing no elements.
Note: ∅ ≠ {∅}

Some examples
• The set V of all vowels in the English alphabet V = {a,
e, i, o, u}.
• The set of positive integers less than 100 can be
denoted by {1, 2, 3, . . . , 99}. ellipses (. . .) are used
when the general pattern of the elements is obvious.
• {a, 2, Fred, New Jersey} is the set containing the four
elements a, 2, Fred, and New Jersey.

Element of Set
• A set is an unordered collection of objects
referred to as elements.
• A set is said to contain its elements. We write
a A to denote that a is an element of the set∈
A.
• The notation a A denotes that a is not an∈
element of the set A.

Try Yourself
• Let A = {1, 3, { { 1,2}, ø},{ø } }. State whether
the following statements are true or not. Give
reason.
• {{1, 2},{ø } } Aϵ
• {1, 4,{ø } } Aϵ
• ø Aϵ

Some Sets
N = {0,1,2,3,…}, the set of natural numbers, non negative integers, (occasionally IN)
Z = { …, -2, -1, 0, 1, 2,3, …), the set of integers
Z+
= {1,2,3,…} set of positive integers
Q = {p/q | p ∈ Z, q ∈Z, and q≠0}, set of rational numbers
R, the set of real numbers
R+, the set of positive real numbers
C, the set of complex numbers.

Set builder notation
• Another way to describe a set is to use set
builder notation.
• O = {x | x is an odd positive integer less than
10}
• or, specifying the universe as the set of
positive integers, as
• O = {x Z+ | x is odd and x < 10}.∈

Empty Set
• There is a special set that has no elements. This set is called the empty
set,or null set, and is denoted by . The empty set can also be denoted by∅
{ }
Common error is to confuse the empty set with the set { }∅ ∅
• The empty set can be thought of as an empty folder and the set consisting
of just the empty set can be thought of as a folder with exactly one folder
inside, namely, the empty folder.
• Determine whether these statements are true or false.
• a) { } b) {∅ ∈ ∅ ∅ ∈ ∅, { }}∅
• c) { } { } d) { } {{ }}∅ ∈ ∅ ∅ ∈ ∅

Subset
• The set A is a subset of B if and only if every
element of A is also an element of B. We use
the notation
A B to indicate that A is a subset of the set B.⊆

Ven diagram of Subset
U
B
A
Fig: A Is a Subset of B.

Try Yourself
• Let A = {1, 5, { { 1,2}, ø},{ø } }. State whether
the following statements are true or not. Give
reason.
• {1, 3, ø} A⊆
• {1, 5, ø} A⊆
• { } A⊆

Proper subset
• When we wish to emphasize that a set A is a
subset of a set B but that
A = B, we write A B and say that A is a proper⊂
subset of B.
∀x(x A → x B) x(x B x A)∈ ∈ ∧ ∃ ∈ ∧ ∈

Try Yourself
A is the set of prime numbers less than 10 , B is the set of odd
numbers less than 10, C is the set of even numbers less than
10.
How many of the following statements are true? Explain
• i. A B⊂ Is ∀x(x A → x B) x(x B x A) true?∈ ∈ ∧ ∃ ∈ ∧ ∈
• ii. B A⊂
• iii. A C⊂
• iv. C A⊂
• v. B C⊂
• vi. C A⊂
• Prime numbers less than 10 are: 2,3,5,7

Set Theory - Definitions and notation
A few more:
Is {a} ⊆ {a}?
Is {a} ∈ {a,{a}}?
Is {a} ⊆ {a,{a}}?
Is {a} ∈ {a}?
Yes
Yes
Yes
No

Power set
• The power set of S is denoted by P(S).
• The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,
• P({0, 1, 2}) = { , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}.∅
• Note that the empty set and the set itself are members of this set of
subsets.

Examples
If S is a set, then the power set of S is 2S
= { x : x ⊆ S }.
If S = {a},
If S = {a,b},
If S = ∅,
If S = {∅,{∅}},
We say, “P(S) is
the set of all
subsets of S.”
2S
= {∅, {a}}.
2S
= {∅, {a}, {b}, {a,b}}.
2S
= {∅}.
2S
= {∅, {∅}, {{∅}}, {∅,{∅}}}.
Fact: if S is finite, |2S
| = 2|S|
. (if |S| = n, |2S
| = 2n
) Why?

Set Theory - Definitions and notation
Quick examples:
{1,2,3} ⊆ {1,2,3,4,5}
{1,2,3} ⊂ {1,2,3,4,5}
Is ∅ ⊆ {1,2,3}?
Yes! ∀x (x ∈ ∅) → (x ∈ {1,2,3}) holds (for all
over empty domain)
Is ∅ ∈ {1,2,3}? No!
Is ∅ ⊆ {∅,1,2,3}? Yes!
Is ∅ ∈ {∅,1,2,3}? Yes!

Set operators
The union of two sets A and B is:
A ∪ B = { x : x ∈ A v x ∈ B}
If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi},
then
A ∪ B = {Charlie, Lucy, Linus, Desi}
A
B

Set Theory -Operators
The intersection of two sets A and B is:
A ∩ B = { x : x ∈ A ∧ x ∈ B}
If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi},
then
A ∩ B = {Lucy}
A
B

The intersection of two sets A and B is:
A ∩ B = { x : x ∈ A ∧ x ∈ B}
20
If A = {x : x is a US president}, and B = {x : x is in this room}, then
A ∩ B = {x : x is a US president in this room} = ∅
A
B
Sets whose
intersection is
empty are called
disjoint sets

The complement of a set A is:
A = { x : x ∉ A}
If A = {x : x is bored}, then
A = {x : x is not bored}
A ∅= U
and
U = ∅
U
I.e., A = U – A, where U is the universal set.
“A set fixed within the framework of a theory and consisting of all objects
considered in the theory. “

Try yourself
• (i) Identify the area ¬ R(x) I(x) ¬ M(x)˄ ˄
• Determine True or False: ∀x R(x) M(x) I(x) → D(x)˄ ˄

Rules of Inference
• We can always use a truth table to show that
an argument form is valid. We do this by
showing that whenever the premises are true,
the conclusion must also be true.

Example
• “If you have a current password, then you can log onto the network.”
“You have a current password.”
• Therefore, “You can log onto the network.”
• We would like to determine whether this is a valid argument.
• That is, we would like to determine whether the conclusion “You can log onto the
network” must be true when the premises “If you have a current password, then
you can log onto the network” and “You have a current password” are both true.
• Use p to represent “You have a current password” and q to represent “You can log
onto the network.”

• Then, the argument has the form
• p → q
• p
• ∴ q
• where is the symbol that denotes “therefore.” the statement∴ ((p → q) ∧
p) → q is a tautology
• This argument is valid because when (premises) p → q and p are both
true, then the (conclusion) q is true.
p q p q
T T T
T F F
F T T
F F T

Try Yourself
• . Verify whether the following argument is valid or not.
a) ￢ q
p → q
∴ ￢ p
b) p → q
q → p
∴ p q˅

Set in discrete mathematics

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