Equal Sets (Set theory part 4 NCERT)

0
734

Equal Sets

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: x2 = 25}, 
E = {x : x is an integral positive root of the equation x2 – 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 n2 ≤ 4} and B = {x : x ∈ R and x2 – 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 in a set do not change a 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.