Types of Relations

In CBSE Class 12, Relations and Functions is worth about 8 marks, and the very first skill tested is classifying a relation. A relation on a set is nothing more than a subset of ordered pairs — once you internalise the three test properties, these become guaranteed marks.

Core definition

A relation R on set A is a subset of A × A. If (a, b) ∈ R we say "a is related to b", written aRb.

Special relations

• Empty relation: R = ∅ — no element of A is related to any element. (R ⊆ A × A, the smallest.)
• Universal relation: R = A × A — every element is related to every element. (The largest.)
• Both empty and universal are called trivial relations.

The three test properties

• Reflexive: (a, a) ∈ R for every a ∈ A.
• Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R.
• Transitive: (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.

Example: on A = {1, 2, 3}, R = {(1,2),(2,3)} is NOT transitive — it has (1,2) and (2,3) but is missing (1,3). Adding (1,3) would fix transitivity.

Exam Tricks & Tips

• 🎯 To DISPROVE a property, one counter-example is enough; to PROVE it, argue for all elements.
• 🎯 Reflexive needs every (a,a) — missing even one diagonal pair kills it.
• 🎯 "aRb if a ≤ b" on real numbers: reflexive ✓, transitive ✓, symmetric ✗ (2 ≤ 3 but 3 ≰ 2).
• 🎯 "aRb if a divides b": reflexive & transitive, not symmetric (2|6 but 6∤2).
• 🎯 Empty relation is symmetric & transitive vacuously, but reflexive only if A = ∅.
• ❌ Common mistake: assuming symmetric + transitive ⇒ reflexive. False — the empty relation on a non-empty set is a counter-example.

Expected exam pattern

1-mark MCQ ("which property fails?") or a 2–3 mark proof asking you to check all three properties for a given R.

Quick recap

Relation = subset of A × A. Learn the three tests cold: reflexive (all diagonals), symmetric (pairs flip), transitive (chains close). Disprove with one counter-example.

Cover the answer, recall, then check. 12 cards on the must-know facts about types of relations.

Q1. What is a relation R on a set A, formally?
A1. Any subset of A × A. A pair (a,b) ∈ R means a is related to b.

Q2. Define the empty relation and universal relation.
A2. Empty: R = ∅ (nothing related). Universal: R = A × A (everything related to everything).

Q3. State the reflexive property.
A3. (a, a) ∈ R for every a ∈ A — every element relates to itself.

Q4. State the symmetric property.
A4. If (a, b) ∈ R then (b, a) ∈ R.

Q5. State the transitive property.
A5. If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

Q6. Is "a ≤ b" on ℝ symmetric? Why?
A6. No. 2 ≤ 3 but 3 ≰ 2. It is reflexive and transitive though.

Q7. Is "a divides b" on ℕ symmetric?
A7. No — 2 divides 6 but 6 does not divide 2. It is reflexive and transitive.

Q8. How do you DISPROVE a property?
A8. Give a single counter-example.

Q9. How many relations are possible on a set with n elements?
A9. 2^(n²), since A × A has n² pairs and each is in or out.

Q10. Does symmetric + transitive imply reflexive?
A10. No. The empty relation on a non-empty set is symmetric and transitive but not reflexive.

Q11. Is the empty relation transitive?
A11. Yes, vacuously — there are no pairs to violate the condition.

Q12. On A = {1,2,3}, is R = {(1,1),(2,2)} reflexive?
A12. No — (3,3) is missing, so reflexivity fails.

A seating chart, a family tree, the "is taller than" comparison in your class — every one of these is secretly a relation, a rule that pairs elements of one set with elements of another. Class 12 formalises this everyday idea and then classifies relations by the special properties they satisfy.

Definition: A relation R from set A to set B is any subset of the Cartesian product A × B. When A = B we call R a relation on A. Writing (a, b) ∈ R (read "a is related to b") just means the ordered pair sits inside that subset.

From pairs to properties

If A has m elements and B has n elements, A × B has mn ordered pairs, so the number of possible relations from A to B is 2^(mn) — every subset is a legal relation. Two extremes anchor the whole topic:

• Empty relation R = φ: no element is related to any element. Example on A = {1,2,3}: R = {(a,b) : a − b = 8}. Since no difference equals 8, R is empty.
• Universal relation R = A × A: every element is related to every element. Example: R = {(a,b) : |a − b| ≥ 0} on any set of reals.

Both empty and universal relations are called trivial relations.

The three defining properties (on a set A)

• Reflexive: (a, a) ∈ R for every a ∈ A. Each element relates to itself. ("=" and "is parallel to" are reflexive.)
• Symmetric: (a, b) ∈ R ⟹ (b, a) ∈ R. Relation reads the same both ways. ("is a sibling of" is symmetric; "is the father of" is not.)
• Transitive: (a, b) ∈ R and (b, c) ∈ R ⟹ (a, c) ∈ R. Relations chain. ("<" is transitive: a < b, b < c ⟹ a < c.)

WHY these three? They are exactly the abstract skeleton of the notion "sameness". A relation that is reflexive, symmetric and transitive is an equivalence relation — the star of the next lesson — because it behaves like equality without literally being equality.

Worked example

Question: On A = {1, 2, 3}, let R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Classify R.

Solution:
Step 1 — Reflexive? Need (1,1), (2,2), (3,3). All three present ⟹ reflexive. ✓
Step 2 — Symmetric? The only "mixed" pairs are (1,2) and (2,1); both present, and the diagonal pairs are self-symmetric ⟹ symmetric. ✓
Step 3 — Transitive? Check chains: (1,2) & (2,1) ⟹ need (1,1) — present. (2,1) & (1,2) ⟹ need (2,2) — present. No chain fails ⟹ transitive. ✓
Conclusion: R is reflexive, symmetric and transitive, hence an equivalence relation.

Exam application

Board and JEE questions almost always ask you to test the three properties or to count relations. Two must-know counts on a set with n elements:

• Number of reflexive relations = 2^(n²−n).
• Number of symmetric relations = 2^(n(n+1)/2).

Shortcut / mnemonic — "RST": Reflexive = Row of self-pairs present; Symmetric = Swap survives; Transitive = Two-step trip allowed. Test in the order R → S → T and stop at the first failure to save time.

Key formulas / definitions

• Relation R on set A: subset of A × A.
• Reflexive: (a, a) ∈ R for all a.
• Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R.
• Transitive: (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R.
• Empty relation: R = ∅; universal relation: R = A × A.
• Number of relations on a set of n elements: 2^(n²).

Relations are classified by which of the reflexive, symmetric and transitive properties they satisfy.

Worked Example

Problem: On the set of real numbers R, define a relation R by "a R b if and only if a ≤ b". Determine whether it is reflexive, symmetric, and transitive.

Solution:
Step 1 — Test reflexivity. A relation is reflexive if a R a for every a. Since a ≤ a is always true, the relation is reflexive.

Step 2 — Test symmetry. It is symmetric if a R b implies b R a. Take a = 2, b = 3: 2 ≤ 3 is true, but 3 ≤ 2 is false. So the relation is NOT symmetric.

Step 3 — Test transitivity. It is transitive if a R b and b R c imply a R c. If a ≤ b and b ≤ c, then a ≤ c is always true. So the relation IS transitive.

Step 4 — Conclude. The relation "≤" is reflexive and transitive but not symmetric (it is in fact a partial-order relation, not an equivalence relation).

Answer: The relation "≤" is reflexive and transitive but not symmetric.