Relation On a Set vs Relation Between Two Sets: What Changes?

Two phrases appear side by side in the chapter on relations and students often mix them up:

• "a relation from A to B" (or "between A and B")
• "a relation on A"

They are not different kinds of object. The second phrase is just the special case of the first where the source and target sets are the same. But that small change unlocks several new properties — reflexive, symmetric, transitive — that would make no sense for general relations between two different sets.

The formal definitions

The change is purely semantic. A relation "on A" is the special case B = A of a relation "from A to B." No new axioms are added; the Cartesian product is still the ambient space; pairs are still ordered. The only thing that is different is that pairs now have both coordinates drawn from a single set.

When A = B, new questions become meaningful

When B is a different set from A, asking "is (a, a) \in R?" is usually nonsense — because the second coordinate is supposed to come from B, and a is an element of A, not B. The pair (a, a) might not even be a valid member of A \times B.

Concrete example. Let A = \{Aarav, Isha, Meera\} and B = \{Mathematics, Physics, Chemistry\}, with R being "is enrolled in." A valid pair is (Aarav, Mathematics). The "pair" (Aarav, Aarav) is not in A \times B at all, because Aarav is not a subject. So reflexivity — the property that every element is related to itself — is undefined for this relation. It cannot be true, it cannot be false; the question does not apply.

Now take a relation on a single set. Let A = \{Aarav, Isha, Meera\} and let R be "is a sibling of." A valid pair is (Aarav, Isha), and also (Aarav, Aarav) is a valid candidate pair — it is in A \times A. Now the question "is Aarav a sibling of Aarav?" at least makes sense; you can argue whether someone is their own sibling (usually no), but the pair is an element of the universe A \times A, so it can be in or out of R.

The properties that need B = A

Three key properties only make sense for relations on a single set. Each of them involves comparing two pair-positions that must be drawn from the same underlying set.

• Reflexive: (a, a) \in R for every a. Requires the same element to be a valid first and second coordinate, so B = A.
• Symmetric: (a, b) \in R implies (b, a) \in R. The swap "b in the first slot and a in the second" requires a to be a valid first coordinate and b a valid second coordinate — only guaranteed when both come from the same set.
• Transitive: (a, b) \in R and (b, c) \in R imply (a, c) \in R. The element b appears as both a second coordinate (in the first pair) and a first coordinate (in the second pair). So b must live in both A and B — again forcing B = A.

For a relation from A to B with A \neq B, all three of these properties are meaningless. You can still talk about domain, range, inverse, and composition — but reflexive / symmetric / transitive are off the table.

Properties that work for any relation

Not every relational concept requires B = A. The following are defined for general relations from A to B.

Concept	Definition	Needs B = A?
Domain of R	First coordinates appearing in R	No
Range of R	Second coordinates appearing in R	No
Inverse R^{-1}	\{(b, a) \mid (a, b) \in R\} (a relation from B to A)	No
Complement	(A \times B) \setminus R	No
Cardinality	$	R
Function	Each a paired with exactly one b	No
Reflexive	(a, a) \in R for all a	Yes
Symmetric	(a, b) \in R \Rightarrow (b, a) \in R	Yes
Transitive	Chain closure	Yes
Equivalence relation	Reflexive + symmetric + transitive	Yes

The bottom four require both coordinates to come from the same set — so they are properties of relations on a set, not of relations from one set to another.

How the phrasing signals what you need to do

On an exam, watch the preposition.

• "R is a relation from A to B ..." or "between A and B": the sets are possibly different. Expect questions about domain, range, inverse, cardinality, or whether R is a function.
• "R is a relation on A ...": the sets are the same. Expect questions about reflexive, symmetric, transitive, equivalence relation, equivalence classes, antisymmetric, partial order.

JEE problems are careful with this phrasing. If a problem asks you to check "whether R is reflexive," you know implicitly that R lives on a single set — even if the problem does not spell out B = A.

Bridging the two: a relation from A to B can still be a relation on a bigger set

Sometimes a relation from A to B gets re-interpreted as a relation on A \cup B — the union of both sets. Inside A \cup B, every pair from A \times B is a pair of elements of A \cup B, so the three special properties can now be asked about. But if the relation only contains pairs with first coordinate in A and second coordinate in B, it cannot be reflexive on A \cup B unless A and B overlap — and in most cases it will fail symmetry too.

This re-interpretation is mostly a trick for special problems. The cleaner way is: relations between two different sets are classified by domain/range/inverse behaviour; relations on a single set are classified by reflexive/symmetric/transitive behaviour.

One-line answer

A relation "from A to B" is any subset of A \times B — the two sets can be different. A relation "on A" is the special case B = A. The difference matters because the properties reflexive, symmetric, and transitive are only defined when both coordinates come from the same set.

Related: Relations · Equivalence Relations · Cartesian Product Grid · A × B vs B × A