Given A = {1, 2, 3} and B = {3, 1, 2}, which statement is correct?
AA is a proper subset of B
BB contains more elements than A
CA and B are equal by extensionality
DA ⊂ B but B ⊄ A
By the principle of extensionality, two sets are equal if and only if they have exactly the same elements. Order and repetition are irrelevant in sets, so {1, 2, 3} and {3, 1, 2} are the same set. A proper subset relationship would require A to have fewer elements than B.
Question 2 True / False
The set {2, 2, 5, 5, 5} contains five elements.
TTrue
FFalse
Answer: False
Sets contain only distinct elements — duplicates collapse to one. {2, 2, 5, 5, 5} is the same set as {2, 5}, which has exactly two elements. This is one of the defining properties that distinguishes sets from sequences or multisets.
Question 3 Short Answer
What is the difference between the statements x ∈ S and A ⊆ B?
Think about your answer, then reveal below.
Model answer: x ∈ S means x is a member (element) of the set S — a relationship between an object and a set. A ⊆ B means every element of A is also an element of B — a relationship between two sets.
Membership (∈) relates an individual object to a set; subset (⊆) relates two sets to each other. A common confusion is treating the subset symbol as membership when the left side is a set itself. For example, {1} ∈ {{1}, 2} (set as element) differs from {1} ⊆ {1, 2} (subset).