Questions: Set Equality and Extensionality

Question 1

Which of the following sets is equal to {x ∈ ℤ : x² < 5}?

• A. {0, 1, 2} — only positive integers whose squares are less than 5
• B. {1, 4} — the squares that are less than 5
• C. {-2, -1, 0, 1, 2} — all integers whose square is less than 5
• D. {x : x < √5} — real numbers less than the square root of 5

Answer: (C)

Extensionality says sets are equal iff they have the same members. The integers x with x² < 5 are exactly -2, -1, 0, 1, 2 (since (-2)² = 4 < 5, but (-3)² = 9 ≥ 5). Option A misses the negative integers. Option B lists the values of x² rather than the values of x. Option D is a different set of real numbers. The description 'x ∈ ℤ such that x² < 5' and the enumeration {-2,-1,0,1,2} are different intensions but identical extensions.

Question 2

A student argues that {x | x is a positive even number less than 10} and {2, 4, 6, 8} are different sets because one is defined by a rule and the other by explicit listing. The student's reasoning is:

• A. Correct — the method of definition is part of a set's identity in formal set theory
• B. Correct — the rule-defined set includes all even numbers less than 10, while the list is finite
• C. Incorrect — by extensionality, sets with identical members are the same set regardless of description
• D. Incorrect — but only because both are finite sets; infinite sets defined by rules can differ from enumerated sets

Answer: (C)

The principle of extensionality states that a set is completely determined by its members, not by the description used to define it. {x | x is a positive even number less than 10} and {2, 4, 6, 8} have exactly the same elements, so they are the same set — not just 'equivalent' or 'interchangeable' but literally identical as mathematical objects. The method of construction (rule vs. list) is an intensional property; extensionality is an axiom that discards it.

Question 3

Two sets can have exactly the same elements but still be different sets if they were constructed using different methods or described differently.

• T. True
• F. False

Answer: False

This is precisely what the axiom of extensionality denies. Sets have no 'identity' beyond their membership roster. {1, 2, 3}, {3, 1, 2}, {x ∈ ℕ : x ≤ 3}, and 'the set of positive integers not greater than 3' all refer to the same object in set theory. The construction method, the notation, and the name are all irrelevant — only the members determine the set. This is what makes set theory a clean foundation for mathematics.

Question 4

The uniqueness of the empty set follows from extensionality: any two sets with no elements must be identical because they have the same (vacuous) membership roster.

• T. True
• F. False

Answer: True

Suppose A and B are both empty sets. For any x: x ∈ A is vacuously false, and x ∈ B is vacuously false. So x ∈ A iff x ∈ B holds trivially for every x. By extensionality, A = B. This means we can speak of 'the' empty set (∅) rather than 'an' empty set — it is a unique, definite mathematical object. The axiom of extensionality is what licenses this definite article.

Question 5

According to the principle of extensionality, what is the only criterion that determines whether two sets are equal, and why does this mean {1, 2, 3} = {3, 1, 2}?

Model answer: Extensionality says two sets are equal if and only if they have exactly the same members — nothing else determines set identity. Order of listing is not a property of a set; sets are unordered collections. To check whether {1,2,3} = {3,1,2}: verify that every element of the first is in the second (1 ∈ {3,1,2}, 2 ∈ {3,1,2}, 3 ∈ {3,1,2} — yes) and every element of the second is in the first (same check, yes). Since membership is identical, they are the same set.

This is often surprising to students accustomed to thinking about ordered sequences. In set theory, {1,2,3} is not a sequence — it is a collection, and collections have no intrinsic ordering. Two notations 'listing' the same elements in different orders are two ways of writing the same thing, just as '2+1' and '1+2' are two ways of writing the number 3. Extensionality formalizes this intuition precisely.