1.5 Equal Sets - Sets - class 11 Mathematics
1.5 Equal Sets
NCERT Class 11 Mathematics Textbook for blind and visually impaired students made screen readable by Professor T K Bansal.
Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal.
Clearly, the two sets have exactly the same elements.
Definition 3
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.
We consider the following examples :
(i) Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B.
(ii) Let A be the set of prime numbers less than 6, and P the set of prime factors of 30. Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.
Note: A set does not change if one or more elements of the set are repeated.
For example, the sets A = {1, 2, 3}, and B = {2, 2, 1, 3, 3}, are equal, since each element of A is in B and vice-versa.
That is why we generally do not repeat any element in describing a set.
Example 7
Find the pairs of equal sets, if any, give reasons:
A = {0},
B = {x : x > 15 and x < 5},
C = {x : x − 5 = 0 },
D = {x: x^2 = 25},
E = {x : x is an integral positive root of the equation \(x^2\ −\ 2x\ −\ 15\ =\ 0\)}.
Solution:
Since 0 ∈ A and 0 does not belong to any of the sets B, C, D and E, it follows that, A ≠ B, A ≠ C, A ≠ D, A ≠ E.
Since B = φ but none of the other sets are empty. Therefore B ≠ C, B ≠ D and B ≠ E.
Also C = {5} but −5 ∈ D, hence C ≠ D.
Since E = {5}, C = E.
Further, D = {−5, 5} and E = {5}, we find that, D ≠ E.
Thus, the only pair of equal sets is C and E.
Example 8
Which of the following pairs of sets are equal? Justify your answer.
(i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”.
(ii) A = {n : n ∈ Z and n^2 ≤ 4} and B = {x : x ∈ R and x^2 − 3x + 2 = 0}.
Solution:
(i) We have,
X = {A, L, L, O, Y},
B = {L, O, Y, A, L}.
Then X and B are equal sets as repetition of elements does not change the set. Thus,
X = {A, L, O, Y} = B
(ii)
A = {−2, −1, 0, 1, 2},
B = {1, 2}.
Since 0 ∈ A and 0 ∉ B, A and B are not equal sets.
EXERCISE 1.2
Question 1.
Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) { x : x is a natural numbers, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}
Answer 1.
(i), (iii), (iv)
Question 2.
Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . . }
(iii) {1, 2, 3, . . .99, 100 }
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Answer 2.
(i) Finite
(ii) Infinite
(iii) Finite
(iv) Infinite
(v) Finite
Question 3.
State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
Answer 3.
(i) Infinite
(ii) Finite
(iii) Infinite
(iv) Finite
(v) Infinite
Question 4.
In the following sets, state whether A = B or not:
(i) A = { a, b, c, d }, B = { d, c, b, a }
(ii) A = { 4, 8, 12, 16 }, B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10 }, B = { x : x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10}, B = { 10, 15, 20, 25, 30, . . . }
Answer 4.
(i) Yes
(ii) No
(iii) Yes
(iv) No
Question 5.
Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x^2 + 5x + 6 = 0}
(ii) A = { x : x is a letter in the word FOLLOW}, B = { y : y is a letter in the word WOLF}
Answer 5.
(i) No
(ii) Yes
Question 6.
From the sets given below, select equal sets :
A = { 2, 4, 8, 12},
B = { 1, 2, 3, 4},
C = { 4, 8, 12, 14},
D = { 3, 1, 4, 2}
E = {−1, 1},
F = { 0, a},
G = { 1, −1 },
H = { 0, 1}
Answer 6.
B = D,
E = G