Set Theory Class 11 CBSE Board by Radius JEE Coaching in Aliganj Lucknow.
- Set Theory Class 11 CBSE Board by Radius JEE is given below.
- Sets are used to define the concepts of relations and functions.
- The study of geometry, sequences, probability, etc. requires the knowledge of sets.
- The theory of sets was developed by German mathematician Georg Cantor (1845-1918)
- We often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc.
Examples of Collection of Object
(i) Odd natural numbers less than 12, i.e., 1, 3, 5, 7, 9, 11
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) Various kinds of Quadrilateral
(v) Prime factors of 45, namely, 3, and 5
(vi) The solution of the equation: x2 – 7x + 12 = 0, viz, 3 and 4.
We note that each of the above example is a well-defined collection of objects
A set is a well-defined collection of objects.
We give below a few more examples of sets used particularly in mathematics, viz.
N : the set of all natural numbers
Z : the set of all integers Q : the set of all rational numbers
R : the set of real numbers Z+ : the set of positive integers
Q+ : Set of positive rational numbers, and
R+ : the set of positive real numbers.
(i) Objects, elements and members of a set are same thing.
(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.
If c is an element of a set A, we say that “c belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write c ∈ A. If ‘k’ is not an element of a set A, we write k ∉ A and read “k does not belong to A”. Thus, in the set V of vowels in the English alphabet, a ∈ V but b ∉ V. In the set P of prime factors of 45, 3 ∈ P but 15 ∉ P
Methods of Representation of a Set
There are two methods of representing a set:
(i) Roster or tabular form
(ii) Set-builder form
(i) Roster or tabular form
(A) 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 9 is described in roster form as {2, 4, 6, 8}. Some more examples of representing a set-in roster form are given below:
(1) The set of all natural numbers which divide 12 is {1, 2, 3, 6, 12}
(2) In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as {1, 3, 2, 12, 6}. But Element must not be repeated.
(3) The set of all vowels in the English alphabet is {a, e, i, o, u}
(4) The set of all odd natural numbers less than 9. Therefore, X = {1, 3, 5, 7}
(5) Set of all letters in the word MATHEMATICS. Therefore, Z = {M, A, T, H, E, I, C, S}
(6) The set of even natural numbers is represented by {2, 4, 6, 8. . .}. The dots tell us that the list of even numbers continue indefinitely.
(7) 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 propert
(ii) Set-builder form
(B) 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
{2,4,6,8}, all the elements possess a common property, namely, each of them
is an even natural number less than 9, and no other number possess this property. Denoting
this set by A, we write
A = {x: x is an even natural number less than 9}
Z = {x: x is a natural number and 4 < x < 12} is read as “the set of all x such that
x is a natural number and x lies between 4 and 12.
Let set of all natural numbers which divide 12 is {1, 2, 3, 4, 6, 12}. Written in Roster form
(Tabular form)
A = {1, 2, 3, 4,6, 12}
We can write Set A in Set builder form as A = {x: x is a natural number which divide 12}
Example 1 Write the solution set of the equation x2 + x – 20 = 0 in roster form.
Solution The given equation can be written as
(x – 4) (x + 5) = 0, i. e., x = 4, – 5
Therefore, the solution set of the given equation can be written in roster form as {4, – 5}.
Example 2 Write the set A = {1, 4, 9, 16, 25, . . . } in set-builder form.
Solution We may write the set A as
Z = {x : x is the square of a natural number}
Alternatively, we can write
Z = {x : x = n2, where n ∈ N
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