Theory of Sets Basic Concepts

The Theory of Sets Basic Concepts

Introduction to the Theory of Sets

  • German mathematician Georg Cantor developed the theory of sets in the end of the 19th century.
  • Sets are used to define the concepts of relations and functions.
  • Sets are used in the study of geometry, sequences, probability, etc.

Definition of Sets

  • A collection is a group of objects of a particular kind.
  • A well-defined collection of objects is called a Set.
  • Any element (a) belongs to set A is represented by a ∈ A.
  • n(A) represents the number of elements in a set A.
Examples of Collections:
  • A group of cricket players.
  • A group of few playing cards.
  • A group of good students.
  • A group of some positive numbers.
Examples of Sets:
  • All the vowels in the English Alphabet.
  • All the prime numbers below 10.
  • All the cricket players with centuries more than 20 in the ODIs.
  • All the blacks colored cards in a pack of playing cards.
  • All the natural numbers (N).
  • All the rivers of India.
  • All kinds of triangles.

Representation of Sets

1. Roster form / Tabular form / List form

  • All the elements of a set are shown as a list enclosed in { }.
  • The elements are separated by commas.
  • The order of the elements does not matter.
  • Repetition of the elements is not allowed.
  • For example, A set of even numbers less than 10 = {2, 4, 6, 8}

 

2. Set-builder form

  •  In the set-builder form, sets are represented by a unique common property of all the elements.
  • Set is defined by the definition of the unique common property.
  • This definition is enclosed by { }.
  • For example, S = {x : x is an even natural number and x < 10}
 

Roster form ⇔ Set-builder form

{ ½, ⅓, ¼, ⅕ } = {x : x = 1/(n+1); where n ∈ N and n ≤ 5}

How to read the Set-builder form of a set?

Let us take a set A as shown:
A = {x : x = 1/(n+1); where n ∈ N and n ≤ 5}Then,
A is a set of all x such that x = 1/(n+1) where n is a natural number equal or less than 5.

Types of Set

Empty Set

  • It does not contain any element in the set.
  • It is also called a void set.
  • The empty set is denoted by the symbol or { }.
  • n(A) = 0.

 

Non-empty Set

  • It contains at least one element in the set.
  • The number of elements can be finite or infinite.
  • n(A) ≠ 0.

Finite Set

  • It contains a definite or finite number of elements.
  • The number of elements can be counted.
  • An empty set is also an example of an empty set.
  • n(A) = finite number.

Infinite Set

  • The number of elements can not be counted.
  • The number of elements is not definite/finite.
  • n(A) ≠ finite number.
  • Infinite sets cannot be described in the roster form.

Equal Sets

  • Two sets A and B are said to be equal if they have exactly the same elements.
  • Equal sets are written as A = B.

Unequal Sets

  • Two sets A and B are called unequal sets if they are not equal sets.
  • Equal sets are written as A ≠ B.

Subset

  • Set (A) is said to be a subset of a set (B) if every element of set (A) is also present in the set (B).
  • A ⊂ B if a ∈ A ⇒ a ∈ B.
  • ‘⇒’ means ‘that implies’.
  • A ⊂ B ⇒ A is a subset of B.
  • A ⊄ B ⇒ A is not a subset of B.
  • (an empty set) is a subset of every set.
  • The set (A) itself is also a subset of A. (A ⊂ A)

Universal Set (U)

  • A universal set is a set of all the elements or objects in any context.
  • Every set in that context is a subset of U.

Power Set – (Set of all the subsets of set A)

  • The collection of all subsets of a set A is called the power set of A.
  • It is denoted by P(A).
  • In P(A), every element is a set.

How to find the number of elements in a Power Set?

>> For example, A = {1, 2}
>> n(A) = 2

>> Subset of A = { }, {1}, {2}. {1,2}
>> P(A) = { { }, {1}, {2}. {1,2} }
>> Number of elements in the Power set of A = n(P(A)) = 4 =  .

 

In general, if n(A) = m then n(P(A)= 2m.

 

Some Important Sets of Numbers

  • Natural Numbers (N) = A set of all-natural numbers.
  • Whole Numbers (W) = A set of zero and all natural numbers.
  • Integers (I) or (Z) = All the whole numbers along with their negatives.
  • Rational Numbers (Q) = All the numbers that can be expressed as a fraction or a quotient (p/q form) is called a Rational Number where (q ≠ 0) and (p & q) both are co-prime numbers i.e. (p & q) don’t have any common factor.
  • Irrational Numbers (T) = All the real numbers that are not rational.
  • Real Numbers (R) = All the rational and irrational numbers.

Examples of subsets

  • N ⊂ W ⇒ N is a subset of W
  • W ⊂ I ⇒ W is a subset of I
  • I ⊂ Q ⇒ I is a subset of Q
  • T ⊂ R ⇒ T is a subset of R
  • Q ⊂ R ⇒ Q is a subset of R
In short, N ⊂ W ⊂ I ⊂ Q ⊂ R.

Intervals as subsets of R

a and b are two real numbers where a < b. Below are the types of intervals.
  • Closed interval = [a, b]
  • Open interval = (a, b)
  • Semi Open interval = (a, b] and [a, b).

How to write intervals in the form of sets?

  • [a, b] is a set of all the real numbers between a and b. (including a and b)
  • (a, b) is a set of all the real numbers between a and b. (excluding a and b)
  • [a, b) is a set of all the real numbers between a and b. (including a but excluding b)
  • (a, b] is a set of all the real numbers between a and b. (excluding a but including b)
Clearly, all these sets are subsets of Set R (the set of all the real numbers).

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