Questions: Set Theory Basics

Question 1

The set A = {1, 2, 3} and the set B = {3, 1, 2}. Which of the following is true?

• A. A ≠ B, because the elements are listed in a different order
• B. A = B, because sets are equal when they contain exactly the same elements regardless of order
• C. A ⊂ B, because A was defined before B
• D. A ≠ B, because sets must be defined in the same way to be equal

Answer: (B)

Set equality depends only on membership, not on the order elements are listed or how the set was defined. Since A and B contain exactly the same elements — 1, 2, and 3 — they are equal. Order is irrelevant in sets (unlike ordered tuples).

Question 2

The set {1, 2, 2, 3} contains four elements, since the number 2 appears twice.

• T. True
• F. False

Answer: False

By definition, a set contains only distinct elements. Repetitions are ignored — {1, 2, 2, 3} is identical to {1, 2, 3}, a three-element set. This is one of the key differences between sets and multisets (bags), which do allow duplicate elements.

Question 3

What is the difference between A ⊆ B and A ⊂ B?

Model answer: A ⊆ B means A is a subset of B (every element of A is in B, and A may equal B). A ⊂ B means A is a proper subset of B (every element of A is in B, but A ≠ B — B has at least one element A does not).

The distinction mirrors ≤ vs. < for numbers. A ⊆ B allows the possibility A = B, while A ⊂ B requires A to be strictly smaller. Every set is a subset of itself (A ⊆ A), but no set is a proper subset of itself.