Skip to content

Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity

Relations and Their Types for CA Foundation

Updated 1 October 2026

A relation from set A to set B is any subset of the Cartesian product A × B, a set of ordered pairs. To solve questions, list the pairs, find domain and range, then test reflexive, symmetric and transitive properties. A relation with all three is an equivalence relation.

Understand Relations and Their Types

An ordered pair (a, b) is a pair where order matters. (2, 3) is not the same as (3, 2). Two ordered pairs are equal only if both first and second elements match.

The Cartesian product A × B is the set of all ordered pairs (a, b) with a in A and b in B. If A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}. In general A × B and B × A are different sets, unless A = B or one of them is empty.

A relation R from A to B is any subset of A × B. A is the starting set and B is the co-domain. The domain is the set of all first elements of the pairs in R. The range is the set of all second elements. The range is always a subset of the co-domain, but it need not be equal to it.

A relation on a set A (from A to A) can have three key properties. Reflexive: every element is related to itself, so (a, a) is in R for every a in A. Symmetric: whenever (a, b) is in R, (b, a) is also in R. Transitive: whenever (a, b) and (b, c) are in R, (a, c) is also in R.

A relation that is reflexive, symmetric and transitive is an equivalence relation. Examples are equality of numbers and 'has the same remainder when divided by 3'. A relation like 'is less than' is transitive but neither reflexive nor symmetric.

Key formulas to remember

Number of elements in a Cartesian product
n(A × B) = n(A) × n(B)
Holds for finite sets. If either set is empty, the product is empty.
Number of relations from A to B
2^(n(A) × n(B))
Each relation is a subset of A × B, so count the subsets.
Reflexive
(a, a) ∈ R for every a ∈ A
Check against the whole set A, not just the elements that appear in R.
Symmetric
(a, b) ∈ R ⇒ (b, a) ∈ R
Every pair needs its reverse. Pairs like (a, a) are automatically fine.
Transitive
(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
If no chain exists, the condition holds trivially.
Equivalence relation
Reflexive + Symmetric + Transitive
All three must hold.
Domain and range
Domain = {a : (a, b) ∈ R}; Range = {b : (a, b) ∈ R}
Range is a subset of the co-domain.

How to solve Relations and Their Types questions

Use this order for any question on relations. It works whether the relation is given as a list of pairs or as a rule.

  1. 1Write the relation as a set of ordered pairs. If it is given as a rule, apply the rule to every element of the set.
  2. 2Read off the domain (first elements) and the range (second elements). Note the co-domain separately.
  3. 3Test reflexive: check that (a, a) is present for every element of the full set A.
  4. 4Test symmetric: for each pair (a, b) with a ≠ b, look for (b, a).
  5. 5Test transitive: for each pair (a, b), look for pairs starting with b, and confirm (a, c) is present.
  6. 6If one test fails, find one counter-example pair and stop checking that property.
  7. 7Call the relation an equivalence relation only if all three tests pass.
  8. 8Match your result with the MCQ options.

Quickest way: Counter-example and diagonal check

When to use it: Use in MCQs where a relation on a small set is listed as pairs, or a rule is given.

  1. For reflexive, count the diagonal pairs (a, a). You need one for every element of the set. Missing one means not reflexive.
  2. For symmetric, scan for a single pair whose reverse is missing. One miss settles it.
  3. For transitive, look at pairs that chain, such as (a, b) and (b, c). Check that (a, c) exists. One miss settles it.
  4. For rules like 'a < b', test with numbers. 'a < b' fails reflexive since a < a is false, fails symmetric, but is transitive.
  5. For counting questions, use n(A) × n(B) and 2 to that power. Do not list pairs.
  6. If a question looks long and you are unsure, skip it. Negative marking of 0.25 marks per wrong answer applies in Paper 3 (Quantitative Aptitude) and in Paper 4 MCQs, so a wrong guess costs you marks there.

Common mistakes in Relations and Their Types

  • Treating A × B and B × A as the same set.

    Students think of pairs as unordered groups.

    Fix: Remember (a, b) ≠ (b, a) in general. Write pairs in the order first set, then second set.

  • Declaring a relation reflexive because the elements that appear in it have (a, a).

    Students check only the elements that show up in the pairs.

    Fix: Reflexive needs (a, a) for every element of the given set A, including elements that appear in no pair.

  • Checking symmetric by finding one pair that has its reverse.

    One matching case feels like proof.

    Fix: Symmetric needs every pair to have its reverse. One missing reverse disproves it.

  • Saying a relation is not transitive when no chains exist.

    Students expect to find a confirming chain.

    Fix: If no pairs (a, b) and (b, c) chain, the condition holds trivially and the relation is transitive.

  • Confusing range with co-domain.

    Both are sets of second elements, so they sound alike.

    Fix: Co-domain is the whole target set B. Range is only the second elements actually used in R.

  • Assuming a relation with both (a, b) and (b, a) for a ≠ b is transitive without checking (a, a) and (b, b).

    Students see the pairs match up as symmetric and assume the relation is well behaved, so they skip the chain check.

    Fix: Chain (a, b) with (b, a): transitivity then needs (a, a). Chain (b, a) with (a, b): it needs (b, b). Check that both are present before calling it transitive.

Worked examples

Example 1

Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2)} on A. Then R is: (a) reflexive and symmetric only (b) reflexive and transitive only (c) symmetric and transitive only (d) an equivalence relation

Show the solution
  1. Reflexive: (1, 1), (2, 2), (3, 3) are all present, so R is reflexive.
  2. Symmetric: (1, 2) is in R but (2, 1) is not, so R is not symmetric.
  3. Transitive: the chains are (1, 1)-(1, 2) giving (1, 2), present; (1, 2)-(2, 2) giving (1, 2), present. Other chains use diagonal pairs and give pairs already in R. So R is transitive.
  4. So R is reflexive and transitive but not symmetric.

Answer: (b) reflexive and transitive only

Example 2

If n(A) = 3 and n(B) = 2, how many different relations from A to B are there? (a) 6 (b) 8 (c) 32 (d) 64

Show the solution
  1. n(A × B) = 3 × 2 = 6.
  2. Each relation is a subset of A × B.
  3. Number of subsets of a 6-element set = 2^6 = 64.

Answer: (d) 64

Example 3

On the set of integers, R is defined by a R b if a − b is divisible by 5. Which statement is true? (a) R is only reflexive (b) R is reflexive and symmetric but not transitive (c) R is an equivalence relation (d) R is symmetric only

Show the solution
  1. Reflexive: a − a = 0, which is divisible by 5. So R is reflexive.
  2. Symmetric: if a − b = 5k, then b − a = −5k, which is also divisible by 5. So R is symmetric.
  3. Transitive: if a − b = 5k and b − c = 5m, then a − c = 5(k + m), divisible by 5. So R is transitive.
  4. All three hold, so R is an equivalence relation.

Answer: (c) R is an equivalence relation

Exam tips

  • Questions are often small sets with 3 or 4 elements. Write the pairs out; it is faster than reasoning in the abstract.
  • Always check the diagonal first for reflexive. It takes seconds and can eliminate options.
  • Memorise how 'a < b', 'a ≤ b', 'a = b' and 'a divides b' behave. They appear often as rule-based relations.
  • For counting questions, use n(A) × n(B) and 2 raised to that power. Watch for arithmetic slips with powers of 2.
  • If you cannot finish the transitive check in under a minute, move on. In Paper 3 (Quantitative Aptitude) and Paper 4 MCQs, each wrong answer loses 0.25 marks.

Practice questions from Sets, Relations and Functions, Limits and Continuity

Relations and Their Types: frequently asked questions

What is the difference between a relation and a function?

A relation is any subset of A × B. A function is a special relation where every element of the domain A is paired with exactly one element of B. So every function is a relation, but not every relation is a function.

How do I check if a relation is reflexive, symmetric and transitive?

List the pairs. For reflexive, look for (a, a) for every element of the set. For symmetric, look for the reverse of each pair. For transitive, check that every chain (a, b), (b, c) has (a, c).

Is the empty relation symmetric and transitive?

Yes, on any set. There are no pairs to violate the conditions, so both hold. It is reflexive only if the set itself is empty.

What is an example of an equivalence relation?

Equality of numbers is one. 'Has the same remainder on division by 5' on integers is another. Both are reflexive, symmetric and transitive.