more elements of the set are repeated. For example, the sets A = { , , } and B = { , , , , } are equal, since each MATHEMATICS element of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set. Example Find the pairs of equal sets, if any, give reasons: A = { }, B = { x : x > and x < }, C = { x : x – = }, D = { x : x = }, E = { x : x is an integral positive root of the equation x – x – = }.
Solution Since ∈ A and 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 = { } but – ∈ D, hence C ≠ D.
Since E = { }, C = E. Further, D = {– , } and E = { }, we find that, D ≠ E. Thus, the only pair of equal sets is C and E. Example Which of the following pairs of sets are equal?
Justify your answer. X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”. A = { n : n ∈ Z and n ≤ } and B = { x : x ∈ R and x – x + = }. 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 = {– , – , , , }, B = { , }. Since ∈ A and ∉