Class 11 Maths Sets Relations and Functions — Samacheer Kalvi Notes

Quick Summary

Sets, Relations and Functions is the foundation of Class 11 Maths. It introduces the language of modern mathematics — sets give a way to group objects, relations describe connections between sets, and functions are the core tool used in calculus, algebra, and beyond. Every subsequent chapter builds on this one.

Key Concepts

Types of Sets

• Empty set (∅): No elements. \|∅\| = 0
• Singleton: Exactly one element
• Finite / Infinite: Countable vs uncountable elements
• Equal sets: Same elements (order doesn't matter)
• Equivalent sets: Same cardinality (same number of elements)
• Universal set (U): Contains all sets under discussion

Set Operations and Laws

Relations

A relation R from set A to set B is a subset of A × B.

• Domain of R: All first elements (x values)
• Range of R: All second elements (y values)
• Inverse relation R⁻¹: Swap each ordered pair — (a, b) → (b, a)

Types of Relations on a Set A

Functions

A function f : A → B assigns each element of A exactly one element of B.

Types of Functions

Composition of Functions

If f : A → B and g : B → C, then (g ∘ f)(x) = g(f(x))

Note: g ∘ f ≠ f ∘ g in general.

Inverse Function

f⁻¹ exists only if f is bijective. If f(a) = b, then f⁻¹(b) = a.

Worked Examples

Example 3: Determine if f(x) = 2x + 3 is bijective. Solution: One-one: f(a) = f(b) ⟹ 2a + 3 = 2b + 3 ⟹ a = b ✓. Onto: for every y ∈ ℝ, x = (y−3)/2 ∈ ℝ ✓. So bijective, f⁻¹(x) = (x − 3)/2.

Important Questions

One Mark

• Define a function. Give one example.
• State De Morgan's laws.
• If \|A\| = 3, what is \|P(A)\|?
• What is the domain of f(x) = 1/(x − 2)?
• Define an equivalence relation.

Short Answer (3 marks)

• If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, find A ∪ B, A ∩ B, A − B, and B − A.
• Verify De Morgan's law for A = {1,2,3}, B = {2,3,4}, U = {1,2,3,4,5}.
• Show that f(x) = x² from ℝ → ℝ is neither one-one nor onto.

Long Answer (5 marks)

• Using the formula \|A ∪ B ∪ C\|, solve: In a class of 60 students, 30 study Maths, 25 study Physics, 20 study Chemistry. 10 study both M and P, 8 study both P and C, 7 study both M and C, and 5 study all three. How many study none?
• Define types of functions with examples. Prove that f : ℝ → ℝ defined by f(x) = 3x − 5 is bijective and find its inverse.

Common Mistakes

• Confusing equal sets (same elements) with equivalent sets (same count).
• Writing A × B when B × A is asked — order matters in Cartesian products.
• Concluding a function is onto without checking Range = Co-domain.
• Forgetting f⁻¹ requires f to be bijective — not all functions have an inverse.
• Misapplying De Morgan's law — the complement goes on each set, union ↔ intersection flips.

Board Exam Tips

• Venn diagrams with 3 sets and cardinality problems appear almost every year as 3-mark or 5-mark questions — practise the inclusion-exclusion formula with numbers.
• "Prove f is bijective" is a standard 5-mark question — always show one-one first, then onto, then state the conclusion.
• Domain questions: check for zero denominator, negative under square root, negative under log.
• JEE/NEET: composition of functions and inverse functions are tested frequently.