Let A = {1, 2} and B = {a, b}. Which of the following is a valid relation from A to B?
A{(1, a, b), (2, a)} — ordered triples formed from A and B
B{(a, 1), (b, 2)} — pairs with B-elements first
C{(1, a), (2, b), (2, a)} — a proper subset of A × B
D{(1, a), (2, b)} only — a relation must pair every element of A with exactly one element of B
A relation from A to B is any subset of A × B — there is no additional requirement. A × B = {(1,a),(1,b),(2,a),(2,b)}, and {(1,a),(2,b),(2,a)} is a valid subset. Option D describes a function (the additional constraint that every element of A has exactly one partner), which is the most important special case of a relation but not the only valid one. Options A and B fail because they misplace the ordering or form triples rather than pairs.
Question 2 Multiple Choice
Which of the following best describes the relationship between the concepts of 'function' and 'relation'?
AFunctions and relations are completely separate concepts with no formal connection
BEvery function is a relation, but not every relation is a function
CEvery relation is a function — they are different names for the same thing
DFunctions are more general than relations because functions can be defined by formulas
A function f: A → B is a relation from A to B with the extra constraint that every element of A appears as a first coordinate in exactly one pair. So functions are a special case of relations — a subset of A × B that satisfies the vertical-line-test condition. Not every relation is a function: the relation {(1,a),(1,b),(2,a)} has 1 paired with two elements, violating the single-output requirement. Framing functions as relations is one of the unifying moves of set-theoretic foundations.
Question 3 True / False
The less-than relationship on real numbers can be formally expressed as a subset of ℝ × ℝ.
TTrue
FFalse
Answer: True
The less-than relation on ℝ is formally the set {(x, y) ∈ ℝ × ℝ : x < y}. Writing 'x < y' is just informal shorthand for saying the pair (x, y) belongs to this set. This is a key point: what we think of as a 'comparison' or 'connection' between two numbers is formalized as a collection of ordered pairs. Every relation — divisibility, 'is a parent of', 'is congruent to mod n' — admits the same treatment.
Question 4 True / False
The Cartesian product A × B and the Cartesian product B × A contain exactly the same ordered pairs.
TTrue
FFalse
Answer: False
Ordered pairs are sensitive to order: (a, b) and (b, a) are different elements unless a = b. So A × B = {(a, b) : a ∈ A, b ∈ B} and B × A = {(b, a) : b ∈ B, a ∈ A} are different sets whenever A ≠ B. For example, if A = {1} and B = {x}, then A × B = {(1, x)} but B × A = {(x, 1)}. The word 'ordered' in 'ordered pair' is doing real mathematical work here.
Question 5 Short Answer
Why does defining a relation as a subset of a Cartesian product unify concepts like 'less than,' 'divides,' and 'is a function of' under a single mathematical framework?
Think about your answer, then reveal below.
Model answer: All three are ways of associating elements from one set with elements of another (or the same) set. By defining a relation as simply any subset of A × B, we capture the common structure: a pair (a, b) is in the relation if and only if a 'stands in the relation to' b. This means divisibility, ordering, and functional assignment all become instances of the same object — a set of ordered pairs — and any theorem proved about relations in general applies to all of them. It also lets us state precisely what properties (reflexivity, symmetry, transitivity) each type of relation has.
This unification is one of the main payoffs of set-theoretic foundations. Instead of treating 'functions,' 'orderings,' and 'equivalences' as conceptually separate kinds of things, we recognize them as subsets of Cartesian products satisfying different combinations of properties. Equivalence relations are reflexive, symmetric, and transitive; partial orders are reflexive, antisymmetric, and transitive; functions add the single-valued condition. The Cartesian product framework is the common language that makes these distinctions precise and comparable.