Relations · step 6 of 13

Relations as subsets of A × B

Formally a relation from A to B is just a subset of A × B — no more structure than that. The domain is the set of first coordinates actually used, the range (or image) the set of second coordinates. Because a relation is a set, all the usual set operations apply, and the number of relations from A to B is 2^(|A|·|B|), since each pair is independently in or out.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

List the pairs explicitlyWhether the relation is given by a rule or a picture, writing the pairs makes domain and range immediate for finite cases.
Domain = first coordinatesdom(R) = {a : ∃b, (a,b) ∈ R}. It is a subset of A and need not equal A.
Range = second coordinatesran(R) = {b : ∃a, (a,b) ∈ R}, a subset of B. The codomain B is given; the range is computed.
Take images of subsetsR[X] = {b : ∃a ∈ X, (a,b) ∈ R}. The range is R[A], the image of the whole domain.

Worked example

Let A = {1,2,3}, B = {p,q,r} and R = {(1,p),(1,q),(3,q)}. Give dom(R), ran(R), R[{1}] and the number of relations from A to B.

First coordinates used: 1 and 3, so dom(R) = {1,3} — note 2 is absent.
Second coordinates used: p and q, so ran(R) = {p,q}; r is unused.
R[{1}] collects partners of 1: {p, q}.
|A × B| = 9, and each of the 9 pairs is independently in or out.

Answer. dom(R) = {1,3}, ran(R) = {p,q}, R[{1}] = {p,q}, and there are 2⁹ = 512 relations from A to B.

Where marks get dropped

These are the specific errors that cost credit on relations as subsets of a × b questions — QED's rubric penalises each of them separately.

Equating the range with the codomain. The codomain is declared; the range is what is actually hit, and they differ whenever some element of B is unused.
Assuming the domain is all of A. Unlike a function, a relation may leave elements of A unrelated to anything.
Forgetting that one element may have several partners. Relations are not functions, so 1 relating to both p and q is perfectly legal.

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Answer in real notation

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Relations as subsets of A × B — frequently asked questions

When is a relation a function?

Exactly when every element of A appears as a first coordinate exactly once — totality plus single-valuedness. R above fails both: 2 is missing and 1 has two partners.

How many relations are there on a set of size n?

2^(n²), since a relation on A is a subset of A × A which has n² elements. For n = 3 that is 512.

Is the empty set a relation?

Yes — it is a subset of A × B, with empty domain and empty range. It is vacuously symmetric, antisymmetric and transitive.

The rest of Relations

Properties of relations, composition, and representations. Each subtopic below has its own method, worked example and mark-losing traps.

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