Relations and Functions — Class 10 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 10 Mathematics, Chapter 1. The language of pairs, relations and functions.

1. About this chapter

This chapter introduces the Cartesian product, relations, and functions — their types, representations, and composition.

2. Ordered pairs, Cartesian product and relations

• Ordered pair: (a, b) — order matters, so (a, b) ≠ (b, a) unless a = b.
• Cartesian product: A × B = { (a, b) : a ∈ A, b ∈ B }. n(A × B) = n(A) × n(B).
• Relation from A to B: any subset of A × B. It has a domain, co-domain and range.

3. Functions and their types

• A function f : A → B assigns to every element of A exactly one element of B.
• Types: one-one (injective), onto (surjective), bijective (both), into, constant, and identity function.
• Counting: if n(A) = p and n(B) = q, then number of relations = 2^(pq) and number of functions = q^p.

4. Representation and composition

• A function can be shown by an arrow diagram, table, set of ordered pairs, or graph.
• Composition: (f ∘ g)(x) = f(g(x)). In general f ∘ g ≠ g ∘ f.
• Graphs: recognise the shapes of linear, quadratic, cubic and reciprocal functions.

5. Worked examples

Example 1. If A = {1, 2} and B = {3, 4}, find A × B and n(A × B). A × B = {(1,3), (1,4), (2,3), (2,4)}; n(A × B) = 2 × 2 = 4.

Example 2. If n(A) = 3 and n(B) = 2, find the number of functions from A to B. q^p = 2³ = 8.

Example 3. If f(x) = 2x + 1 and g(x) = x², find (f ∘ g)(x). (f ∘ g)(x) = f(g(x)) = f(x²) = 2x² + 1.

6. Common mistakes

• Mistake: Treating any relation as a function. Fix: A function needs exactly one image for every domain element.
• Mistake: Writing (f ∘ g) = (g ∘ f). Fix: Composition is generally not commutative.
• Mistake: Swapping the relation/function counting formulas. Fix: Relations = 2^(pq); functions = q^p.

7. Practice (book-back style)

1. If A = {a, b} and B = {1, 2, 3}, find n(A × B).
2. Define a one-one and an onto function.
3. If n(A) = 2, n(B) = 3, find the number of relations from A to B.
4. If f(x) = x + 3 and g(x) = 2x, find (g ∘ f)(x).
5. State whether (f ∘ g) = (g ∘ f) in general.

8. Answer key

1. n(A × B) = 2 × 3 = 6.
2. One-one: distinct elements have distinct images; onto: every element of the co-domain has a pre-image.
3. 2^(pq) = 2^(6) = 64.
4. (g ∘ f)(x) = g(f(x)) = g(x + 3) = 2(x + 3) = 2x + 6.
5. No — composition is not commutative in general.

9. Quick revision

• Chapter 1 · Cartesian product, relations, functions.
• n(A × B) = n(A)·n(B); relations = 2^(pq); functions = q^p.
• Function: every domain element → exactly one image.
• Types: one-one, onto, bijective, into, constant, identity.
• (f ∘ g)(x) = f(g(x)); not commutative.