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.
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 = 2².
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).
