Equality vs. repetition — identify which pair is not equal: Which pairs of sets below are not equal? (Remember duplicates do not change a set.)

Introduction / Context:
Two sets are equal if they have exactly the same elements, ignoring order and repetitions. We test each pair accordingly.

Given Data / Assumptions:

• (a) A = {1,3}, B = {1,4}
• (b) A = {x : x + 2 = 2} → {0}; B = {0}
• (c) A = {1,3,4}; B = {3,1,4}
• (d) A = {1, 1/2, 1/3, ...}; B = {1/n : n ∈ N}

Concept / Approach:
Reduce duplicates, then compare elements one-to-one. For (d), both describe the same infinite set.

Step-by-Step Solution:
(a) {1,3} ≠ {1,4} → not equal(b) {0} = {0} → equal(c) {1,3,4} = {1,3,4} → equal(d) identical descriptions → equal

Verification / Alternative check:
For (b) solve x + 2 = 2 → x = 0 confirming equality.

Why Other Options Are Wrong:
They assert non-inequality when pairs are equal; only pair (a) differs.

Common Pitfalls:
Letting repetition or order mislead; sets ignore both.

Final Answer:
(a) A = {1, 3, 3, 1}, B = {1, 4}