Discrete Structures: Sets and Their Properties, Slides of Computer Science

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CSC102 - Discrete Structures By Mahwish Waqas

Department Of Computer Science, CUI

Lahore Campus

Course Outline

• Sets
  • Set Terminologies
  • Sets of sets
  • Power Set
  • Cartesian Product
  • Set notation with Quantifier

Set

• A set is an unordered collection of objects.
• The objects in a set are called the elements , or members , of the set.
• A set is said to contain its elements. Example:
  • Z is the set of integers.
  • Cities in the Pakistan: {Lahore, Karachi, Islamabad, … }
  • Sets can contain non-related elements: {3, a, red, Gilgit } Properties:
• Order does not matter
  • {1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}
• Sets do not have duplicate elements
  • Consider the list of students in this class
  • It does not make sense to list somebody twice

Set Membership

• a is an element of the set A , denoted by a ∈ A.
• a is not an element of the set A , denoted by a ∉

A.

Sets (example)

• Example: V: {a,e,i,o,u} a ∈ V b ∉ V I: {0,1,2,…,99} 50 ∈ I 100 ∉ I S: {a,2,class} 2 ∈ S room ∉ S

Specifying a Set

• Capital letters (A, B, S…) for sets
• Italic lower-case letter for elements ( a, x, y …)
• Easiest way: list all the elements
  • A = { 1 , 2 , 3 , 4 , 5 }, Not always feasible!
• May use ellipsis (…): B = { 0 , 1 , 2 , 3 , …}
• May cause confusion. C = { 3 , 5 , 7 , …}. What’s next?
• If the set is all odd integers greater than 2 , it is 9
• If the set is all prime numbers greater than 2 , it is 11

Important Sets

• Set of natural numbers
  • 𝐍 = { 1 , 2 , 3 ,…}
• Set of integers
  • 𝐙 = {…,- 2 ,- 1 , 0 , 1 , 2 ,…}
• Set of positive integers
  • 𝐙
    • = { 1 , 2 , 3 ,…}
• Set of rational numbers
  • 𝐐 = {p/q | p ∈ 𝐙, q ∈ 𝐙, and q ≠ 0 }
• Set of real numbers
  • 𝐑

Examples

• S 1 = { N, Z, Q, R }

• S 1 has 4 elements, each of which is a set.
• S 2 = {x | x ∈ N and ∃k k ∈ N , x = k 2 }
• Set of squares of natural numbers

Equality of Sets (examples)

• { 1 , 2 , 3 } and { 3 , 2 , 1 } { 1 , 2 , 3 } = { 3 , 2 , 1 }
• 𝐙
  • and { 0 , 1 , 2 ,…} 𝐙
    • ≠ { 0 , 1 , 2 ,…}

The Universal Set

• U is the universal set – the set containing all objects or elements (or the “universe”), and of which all other sets are subsets
• For the set {- 2 , 0. 4 , 2 }, U would be the real numbers
• For the set { 0 , 1 , 2 }, U could be the N, Z, Q, R depending on the context
• For the set of the vowels of the alphabet, U would be all the letters of the alphabet

Empty Set (example)

• Example:
• S = {x | x ∈ 𝑍
  • and x < 0 } S = { } = Ø
• A set that has no elements called empty set , or null set.
• Ø and {Ø} Ø ≠ {Ø}

Sets Of Sets

• Sets can contain other sets
• S = { { 1 }, { 2 }, { 3 } }
• T = { { 1 }, {{ 2 }}, {{{ 3 }}} }
• V = { { { 1 }, {{ 2 }} }, { {{ 3 }} }, { { 1 }, {{ 2 }}, {{{ 3 }}} } } V has only 3 elements!
• Note that 1 ≠ { 1 } ≠ {{ 1 }} ≠ {{{ 1 }}}
• They are all different

Subset and Equality

• A ⊆ B, ∀ x (x ∈ A → x ∈ B) and
• B ⊆ A, ∀ x (x ∈ B → x ∈ A) then
• A = B, ∀ x (x ∈ A ↔ x ∈ B)

Subset (example)

• Q and R Q ⊆ R
• N and Z N ⊆ Z
• A = {x | x ∈ 𝐙
  • and x< 10 } B = {x | x ∈ 𝐙
    • , x is even and x< 10 } B ⊆ A