Sets and Functions – Complete Notes for Competitive Exams

1. Introduction

Sets and Functions are fundamental concepts in mathematics and frequently appear in Quantitative Aptitude sections of SSC, Banking, Railways, and other competitive exams.

• Set: A collection of well-defined objects.
• Function: A relation between two sets where each element of the first set corresponds to exactly one element of the second set.

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2. Sets

Definitions

1. Empty set: $\emptyset$∅ or $\{\}${}
2. Finite set: Contains a limited number of elements
3. Infinite set: Contains unlimited elements
4. Subset: $A \subseteq B$A⊆B → All elements of A are in B
5. Proper subset: $A \subset B$A⊂B → A is a subset of B but not equal to B
6. Universal set (U): The set containing all possible elements under consideration

Operations on Sets

• Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$A∪B={x:x∈A or x∈B}
• Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$A∩B={x:x∈A and x∈B}
• Difference: $A – B = \{x : x \in A \text{ and } x \notin B\}$A−B={x:x∈A and x∈/B}
• Complement: $A’ = U – A$A′=U−A

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3. Functions

Definitions

• A function f from set A to set B is denoted $f : A \to B$f:A→B.
• Each element of A has exactly one image in B.

Types of Functions

1. One-to-One (Injective): Different elements in A map to different elements in B
2. Onto (Surjective): Every element of B has at least one pre-image in A
3. Bijective: Both injective and surjective
4. Constant Function: Every element of A maps to the same element in B
5. Identity Function: $f(x) = x$f(x)=x for all $x \in A$x∈A

Important Formulas

• Number of subsets of a set with n elements: $2^n$2n
• Number of one-to-one functions from set A (m elements) to B (n elements): $n(n-1)(n-2)…(n-m+1)$n(n−1)(n−2)…(n−m+1) if $n \ge m$n≥m

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4. Important Tips

• Use Venn diagrams for solving set problems visually
• For functions, check domain and range carefully
• Complement rule: $n(U) = n(A) + n(A’)$n(U)=n(A)+n(A′)
• Focus on injective, surjective, and bijective definitions for exams

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Top 25 Practice Questions – Sets & Functions

Sets

Q1. Find $A \cup B$A∪B if $A = \{1, 2, 3\}, B = \{2, 3, 4\}$A={1,2,3},B={2,3,4}
Q2. Find $A \cap B$A∩B for the same sets
Q3. Find $A – B$A−B
Q4. If $U = \{1,2,3,4,5\}, A = \{2,3\}$U={1,2,3,4,5},A={2,3}, find $A’$A′
Q5. Number of subsets of a set with 5 elements
Q6. Number of proper subsets of a set with 4 elements
Q7. If $A = \{1,2,3\}, B = \{2,3,4,5\}$A={1,2,3},B={2,3,4,5}, find $|A \cup B|$∣A∪B∣
Q8. If $A \subseteq B$A⊆B and $|A| = 3, |B| = 5$∣A∣=3,∣B∣=5, how many subsets of B contain A?
Q9. Solve using Venn diagram: In a class of 50 students, 30 like Maths, 25 like Science, 10 like both. How many like neither?
Q10. If $A = \{1,2,3,4\}, B = \{3,4,5,6\}$A={1,2,3,4},B={3,4,5,6}, find $A \Delta B$AΔB (symmetric difference)

Functions

Q11. Determine if $f(x) = 2x + 3$f(x)=2x+3 is one-to-one
Q12. Determine if $f(x) = x^2$f(x)=x2 is onto from $\mathbb{R} \to \mathbb{R}$R→R
Q13. Find the range of $f(x) = x^2 + 1$f(x)=x2+1
Q14. Is $f(x) = 3x + 7$f(x)=3x+7 bijective if domain and codomain are $\mathbb{R}$R?
Q15. Number of functions from a set with 3 elements to a set with 4 elements
Q16. Number of one-to-one functions from a set with 2 elements to a set with 5 elements
Q17. Find $f(3)$f(3) if $f(x) = x^2 – x + 2$f(x)=x2−x+2
Q18. If $f(x) = x + 2$f(x)=x+2 and $g(x) = 3x$g(x)=3x, find $(g \circ f)(x)$(g∘f)(x)
Q19. Determine if $f(x) = x^3$f(x)=x3 is one-to-one and onto $\mathbb{R} \to \mathbb{R}$R→R
Q20. Domain of $f(x) = \frac{1}{x-2}$f(x)=x−21​
Q21. Range of $f(x) = \sqrt{x-1}$f(x)=x−1​
Q22. If $f(x) = x^2 – 4x + 3$f(x)=x2−4x+3, find $f(0)$f(0) and $f(3)$f(3)
Q23. If $f(x) = x + 1$f(x)=x+1, find $f^{-1}(x)$f−1(x)
Q24. Determine whether $f(x) = x^2$f(x)=x2 is injective on $[0, \infty)$[0,∞)
Q25. If $f(x) = 2x + 1$f(x)=2x+1, $g(x) = x^2$g(x)=x2, find $(f \circ g)(2)$(f∘g)(2)

Answer

Answers – Sets & Functions

Sets Answers

Q1. $\{1,2,3,4\}${1,2,3,4}
Q2. $\{2,3\}${2,3}
Q3. $\{1\}${1}
Q4. $\{1,4,5\}${1,4,5}
Q5. 32
Q6. 15
Q7. 5
Q8. 4
Q9. 5
Q10. $\{1,2,5,6\}${1,2,5,6}

Functions Answers

Q11. Yes
Q12. No
Q13. $[1, \infty)$[1,∞)
Q14. Yes
Q15. $4^3 = 64$43=64
Q16. $5 \times 4 = 20$5×4=20
Q17. 8
Q18. $3(x + 2) = 3x + 6$3(x+2)=3x+6
Q19. Yes, Yes
Q20. $x \in \mathbb{R}, x \neq 2$x∈R,x=2
Q21. $[0, \infty)$[0,∞)
Q22. $f(0) = 3, f(3) = 0$f(0)=3,f(3)=0
Q23. $f^{-1}(x) = x – 1$f−1(x)=x−1
Q24. Yes
Q25. $f(g(2)) = f(4) = 9$f(g(2))=f(4)=9