Set Theory

What is a set?

A set is a well defined any collection of objects (elements).
Use capital letters (A, B, C…...) to denote sets and note within a curly bracket.

EX:   A = Odd numbers less than 10 A = { 1, 3, 5, 7, 9 }
        A = { x such that x is odd and x is less than 10 } A = { x; x is odd, x<10 }

Elements of a Set (∈)

Each object in a set is called an element of the set. It is denoted by lower race letters (a, b, c….)

EX: Assume that, B = {a, e, i, o, u}

a ∈ B

o ∈ B
e ∈ B

u ∈ B
i ∈ B

Sub Sets

A sub set is a set whose elements are all members of another set.

Subset ⊆

Not a Subset ⊆/

Empty set and the whole set are also a subset of any given set.


EX:  A = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
     B = { 2, 4, 6, 8 }
     C = { -4, -3, -2, -1 }
                     B ⊆ A
                     C ⊆/  A

If A is a set with “n” elements then A has 2𝑛 subsets.

A = { 2, 4, 6, 8 }

2𝑛 = 16 subsets

If A is a subset of B, but A is not exactly equal to B. We calledas a proper subset of B.

A ⊆ B and A = B

A = { 2, 4, 6 }

B = { 2, 4, 6, 8 }

A is a proper subset of B

Null Set (Ø / { } )

A set which contains no elements is called null set.

EX: A = Natural number between 9 and 10 A = Ø A = { }

Universal set ( S / U / Ꜫ )

The set containing all objects or elements.

The totality of all the elements in two or more given sets.

EX: If we toss a die, the set of all possible outcomes is the universe.
        S = { 1, 2, 3, 4, 5, 6 }

Disjoint set

A set of sets is disjoint if no pair of those sets has elements in common.

EX: A = { 2, 4, 6, 8 }
    B = { 1, 3, 5, 7, 9 }

Venn Diagrams

A universe can be represented geometrically by the set of point inside a rectangular. In such case subsets are represented by set of points inside circles. Such diagrams are called Venn diagrams.

Finite set and Infinite set

In an infinite set all the members of the set can be listed or a set with a limited number of elements.

Otherwise that set is infinite. Empty set is also finite.

EX: A = { Monday,….., Sunday } → Finite B = { Odd numbers} → Infinite

Set Operations

Union ( U )

The set of all elements which belongs to either A or B or both A and B is called the Union of A and B. (AUB)

EX: A = { 1, 3, 5 }
    AUB = { 1, 2, 3, 4, 5 }
    B = { 2, 4 }

Intersection ( Ո )

The intersection of two sets are those elements that belong to both sets.

EX: A = { 1, 3, 5, 7, 9 }
    B = { 2, 3, 5, 6 }
    AՈB = { 3, 5 }

Difference

Let A and B be two sets. The difference of A and B written as A-B, is the set of all those elements of A which

do not belong to B.

EX: A = { 1, 2, 3, 4 }
    B = { 3, 4, 5, 6 }
    A-B = { 1, 2 }
    B-A = { 5,6}

Complement Complement of set A denoted by A', is the set of all elements in the universal set that are not in A.

If B is a sub set of A, then A-B is collect the complement of B relative to A. It is denoted by B’A.

The complement of AUB is denoted by (AUB)’

Some Theorem Related to Sets

01. AՈA = A AUA =A
02. (AUB)UC = AU(BUC) = AUBUC (AՈB)ՈC = AՈ(BՈC) = AՈBՈC
03. (AUB) = (BUA)
04. AU(BՈC) = (AUB) Ո (AUC)
AՈ(BUC) = (AՈB) U (AՈC)
05. AUØ = A
AՈØ = Ø

06. AUS = S AՈS = A
07. AUA' = S
AՈA' = Ø 08. (A')' =A
S' = Ø Ø = S'
09. (AUB)' = A'ՈB' (AՈB)' = A'UB’

Tutorial Created by K.S. Madhushani & K.N. Manchanayaka