Relations

What is a relation?

In plain words: A × B lists every possible pairing of an element of A with an element of B, and a relation keeps only the pairs that obey a rule. NCERT’s example (page 28) takes P = {a, b, c} and Q = {Ali, Bhanu, Binoy, Chandra, Divya}. P × Q has 3 × 5 = 15 pairs, and the rule “x is the first letter of the name y” keeps four of them:

R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)}.

Here b has two images, and Divya is the image of nothing, because no element of P is the letter d. A relation allows both. A function does not allow the first, since each element of the first set must have exactly one image, but it does allow the second.

Three ways to show a relation

A relation can be written in roster form, in set-builder form, or drawn as an arrow diagram, with an arrow from each element to each of its images (NCERT page 28). For A = {1, 2, 3, 4, 5, 6} and the relation from A to A given by y = x + 1 (NCERT Example 7, page 28):

• set-builder form: R = {(x, y) : y = x + 1}, with x, y ∈ A;
• roster form: R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)};
• arrow diagram: the figure below.

Domain, codomain and range

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

NCERT Mathematics (Class 11), Section 2.3, page 28

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊂ codomain.

NCERT Mathematics (Class 11), Section 2.3, page 28

For y = x + 1 on A = {1, 2, 3, 4, 5, 6}: the domain is {1, 2, 3, 4, 5}, since 6 has no image in A; the range is {2, 3, 4, 5, 6}; and the codomain is all of A, {1, 2, 3, 4, 5, 6} (NCERT Example 7, page 29, which prints a stray comma after the 5 in the domain). The domain need not be all of A, and the range need not be all of B.

NCERT Example 8 (page 29) gives an arrow diagram from P = {9, 4, 25} to Q = {5, 3, 2, 1, −2, −3, −5} in which each number points to both of its square roots. The rule is “x is the square of y”:

• set-builder form: R = {(x, y) : x is the square of y, x ∈ P, y ∈ Q};
• roster form: R = {(9, 3), (9, −3), (4, 2), (4, −2), (25, 5), (25, −5)};
• domain {4, 9, 25}, range {−2, 2, −3, 3, −5, 5}, codomain Q.

The element 1 of Q is not the square root of anything in P, so it is in the codomain but not in the range.

Number of relations between two sets

The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A × B.

NCERT Mathematics (Class 11), Section 2.3, page 29

If n(A) = p and n(B) = q, then A × B has pq elements (see counting a Cartesian product), and a set with pq elements has 2^pq subsets (see power set). Each pair is either in the relation or not, two choices for each of the pq pairs.

For A = {1, 2} and B = {3, 4}, A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} has 4 pairs, so there are 2⁴ = 16 relations from A to B (NCERT Example 9, page 29). The count includes the empty relation ϕ and the whole of A × B.

Relations on a set

A relation R from A to A is also stated as a relation on A.

NCERT Mathematics (Class 11), Section 2.3, page 29

A relation on A is a subset of A × A, so a set with n elements has 2^n² relations on it: 2⁹ = 512 when n = 3.

NCERT Example 19 (page 39) looks at the relation R = {(a, b) : a, b ∈ Q and a − b ∈ Z} on the rational numbers and shows three facts:

• (a, a) ∈ R for every a, because a − a = 0 ∈ Z;
• if (a, b) ∈ R then (b, a) ∈ R, because b − a = −(a − b) is also an integer;
• if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, because a − c = (a − b) + (b − c) is a sum of integers.

These three properties are called reflexive, symmetric and transitive, and a relation with all three is an equivalence relation. The names are studied in Class 12, and the JEE Main syllabus lists types of relations and equivalence relations in Unit 1.

Solved examples

Common mistakes to avoid

Quick revision summary

• A relation R from A to B is a subset of A × B; if (x, y) ∈ R, y is an image of x.
• Domain = set of first entries; range = set of second entries; codomain = B; range ⊂ codomain.
• If n(A) = p and n(B) = q, there are 2^pq relations from A to B; a relation from A to A is a relation on A.

Questions students ask

What is the difference between a relation and the Cartesian product A × B?

The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, and a relation from A to B is any subset of it, usually the pairs that obey a rule. So every relation R from A to B satisfies ϕ ⊂ R ⊂ A × B: ϕ is the smallest relation from A to B and A × B itself the largest. For A = {1, 4} and B = {2, 3, 5}, A × B has 2 × 3 = 6 pairs, and the relation “x is less than y” keeps four of them: (1, 2), (1, 3), (1, 5) and (4, 5).

What are the types of relations in maths?

The main types of relations are the empty relation ϕ, the universal relation A × A, and reflexive, symmetric and transitive relations; a relation with all three properties is an equivalence relation. A relation R on A is reflexive if (a, a) ∈ R for every a ∈ A, symmetric if (a, b) ∈ R always gives (b, a) ∈ R, and transitive if (a, b) ∈ R and (b, c) ∈ R always give (a, c) ∈ R. For example, “is less than” on N is transitive but neither reflexive nor symmetric, while “was born in the same month as” is an equivalence relation. These names are taught in Class 12 and are in JEE Main Unit 1.

How many symmetric relations are there on a set with n elements?

A set with n elements has 2^{n(n + 1)/2} symmetric relations on it. Each of the n pairs (a, a) may be in or out, but the other pairs come in n(n − 1)/2 mirror couples, (a, b) with (b, a), that must be taken or left together, giving n(n + 1)/2 free choices. For a set of 4 elements that is 2¹⁰ = 1024 symmetric relations out of 2¹⁶ = 65 536 in all; if it must also be reflexive, every (a, a) is in and only the couples are free, giving 2^{n(n − 1)/2}, here 2⁶ = 64. Symmetric and reflexive relations are Class 12 topics, in JEE Main Unit 1.

Is the number of relations from A to B the same as from B to A?

Yes, there are as many relations from B to A as from A to B: if n(A) = p and n(B) = q, both A × B and B × A have pq pairs, so each direction has 2^pq relations. The relations themselves need not be the same: a relation from A to B has its first entries in A and one from B to A has them in B. Some can coincide, such as the empty relation; if A = B, the two collections are identical. Reversing every pair of a relation from A to B gives one from B to A (often called its inverse), and this matches the two collections one to one. For n(A) = 2 and n(B) = 4 there are 2⁸ = 256 relations in each direction.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Define R on N by R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈ N}. Write R in roster form and give its domain and range (NCERT Exercise 2.2, Q2).

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   x = 1, 2, 3, so R = {(1, 6), (2, 7), (3, 8)}; domain {1, 2, 3}; range {6, 7, 8}.

2. Q2. Find the domain and range of R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}} (NCERT Exercise 2.2, Q6).

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   R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}; domain {0, 1, 2, 3, 4, 5}; range {5, 6, 7, 8, 9, 10}.

3. Q3. Write R = {(x, x³) : x is a prime number less than 10} in roster form (NCERT Exercise 2.2, Q7).

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   The primes below 10 are 2, 3, 5, 7: R = {(2, 8), (3, 27), (5, 125), (7, 343)}.

4. Q4. Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B (NCERT Exercise 2.2, Q8).

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   n(A × B) = 3 × 2 = 6, so there are 2⁶ = 64 relations.