In math a set is a well-defined collection of objects
Example of sets in math
- Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
- The rivers of India
- The vowels in the English alphabet, namely, a, e, i, o, u
- Various kinds of triangles
Representations of sets in math
There are two methods of representing a set :
- Roster or tabular form
- Set-builder form
Roster or tabular form
In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }. For example, the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}.
More Example of Roaster form of set
- The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}
In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as {1, 3, 7, 21, 2, 6, 14, 42}.
- The set of all vowels in the English alphabet is {a, e, i, o, u}
- The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots tell us that the list of odd numbers continue indefinitely
It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct. For example, the set of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} or {H, O, L, C, S}. Here, the order of listing elements has no relevance.
set-builder form
In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by V, we write
V = {x : x is a vowel in English alphabet}
It may be observed that we describe the element of the set by using a symbol x (any other symbol like the letters y, z, etc. could be used) which is followed by a colon “ : ”. After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces. The above description of the set V is read as “the set of all x such that x is a vowel of the English alphabet”. In this description the braces stand for “the set of all”, the colon stands for “such that”. For example, the set
A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that x is a natural number and x lies between 3 and 10.” Hence, the numbers 4, 5, 6, 7, 8 and 9 are the elements of the set A.
Example of sets in math
Example 1 Write the solution set of the equation x2 + x – 2 = 0 in roster form
Solution The given equation can be written as
(x – 1) (x + 2) = 0, i. e., x = 1, – 2
Therefore, the solution set of the given equation can be written in roster form as {1, – 2}.
Example 3 Write the set A = {1, 4, 9, 16, 25, . . . }in set-builder form.
Solution We may write the set A as
A = {x : x is the square of a natural number}
Alternatively, we can write
A = {x : x = n2 , where n ∈ N}
Types of sets
The Empty Set
A set which does not contain any element is called the empty set or the null set or the void set
Example
B = {x : x2 – 2 = 0 and x is rational number}. Then B is the empty set because the equation x2 – 2 = 0 is not satisfied by any rational value of x
Finite and Infinite Sets
A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite
Example
- Let W be the set of the days of the week. Then W is finite
- Let G be the set of points on a line. Then G is infinite
For example, {1, 2, 3 . . .} is the set of natural numbers, {1, 3, 5, 7, . . .} is the set of odd natural numbers, {. . .,–3, –2, –1, 0,1, 2 ,3, . . .} is the set of integers. All these sets are infinite
All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern
Equal Sets
Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ≠ B
Example
Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B
Subsets
A set A is said to be a subset of a set B if every element of A is also an element of B
In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol “⇒” which means implies. Using this symbol, we can write the definiton of subset as follows:
In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol “⇒” which means implies. Using this symbol, we can write the definition of subset as follows:
We read the above statement as “A is a subset of B if a is an element of A implies that a is also an element of B”. If A is not a subset of B, we write A ⊄ B.
Example
Let A = {1, 3, 5} and B = {x : x is an odd natural number less than 6}. Then A ⊂ B and B ⊂ A and hence A = B
Power Set
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
if A = { 1, 2 }, then
P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}
if A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2m
Universal Set
while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc
For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers
Problems on Union and Intersection of Two Sets
Let A and B be finite sets. If A ∩ B = φ, then
- n ( A ∪ B ) = n ( A ) + n ( B )
- n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )
- n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )