Sets — Class 11 MCQs with Answers

Sets — important questions & MCQs with answers (Class 11 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Which of the following is a well-defined set?

   ACollection of beautiful flowersBCollection of all even integersCCollection of clever studentsDCollection of tall people
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   Answer: Collection of all even integers

   Explanation: A set must be well-defined — every object's membership decidable without opinion, like "even integers." "Beautiful" or "clever" depends on judgment, so those collections aren't sets at all.

2. Q2 of 6Easy

   If A = {1, 2}, the number of subsets of A is:

   A4B3C2D5
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   Answer: 4

   Explanation: A set with n elements has 2ⁿ subsets; here n=2 gives 2²=4: ∅, {1}, {2}, {1,2}. Forgetting to count ∅ and the whole set itself is the usual undercount.

3. Q3 of 6Medium

   Set of all real numbers x with 0 ≤ x < 5 is written as:

   A[0, 5)B(0, 5)C[0, 5]D(0, 5]
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   Answer: [0, 5)

   Explanation: 0≤x<5 includes the left endpoint but excludes the right one, written [0,5) — square bracket for included, round for excluded. Using (0,5) instead wrongly excludes 0 too.

4. Q4 of 6Medium

   Assertion (A): If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}, then A – B = B – A. Reason (R): A – B = {x : x ∈ A and x ∉ B}.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: Using R, A – B = {1, 3, 5} and B – A = {8}, so A – B ≠ B – A and A is false. R is the correct definition of the difference of two sets, so R is true.

5. Q5 of 6Hard

   In a class of 100 students, 50 know English, 40 know Hindi, 35 know French, 15 know English & Hindi, 10 Hindi & French, 8 English & French, 5 all three. Number who know none:

   

   A45B20C30D3
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   Answer: 3

   Explanation: Three-set inclusion-exclusion: |E∪H∪F|=50+40+35−15−10−8+5=97 know at least one language; none = 100−97=3. Skipping the +5 for the triple overlap undercounts and inflates "none."

6. Q6 of 6Hard

   Assertion (A): If A = {x : x ∈ R and x² – 8x + 12 = 0} and D = {6}, then D ⊂ A. Reason (R): For the same set A, the set B = {2, 4, 6} is a subset of A.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is true but R is false

   Explanation: x² – 8x + 12 = (x – 2)(x – 6) = 0 gives A = {2, 6}. Since 6 ∈ A, D ⊂ A and A is true. R is false because 4 ∈ B but 4 ∉ A.

Sets notes: key concepts (Class 11 Mathematics)

Sets are the language of mathematics: well-defined collections of distinct objects. This chapter builds representations (roster and set-builder), classifies sets by size and relation, introduces the power set and universal set, and develops union, intersection, difference and complement with De Morgan's laws and counting.

Set & representation

A well-defined collection of distinct objects. Roster form lists elements in braces; set-builder form states a defining property, e.g. {x : x is even}.

Empty set

The set with no elements, written ∅ or { }. It is a subset of every set; note {∅} and {0} are non-empty, so ∅ is not the same as {∅}.

Finite and infinite sets

A finite set has a definite number of elements, so counting them comes to an end; an infinite set never terminates, like ℕ. Cardinality n(A) counts a finite set.

Equal & equivalent sets

Two sets are equal if they have exactly the same elements, regardless of order or repetition; they are equivalent if they merely share the same cardinality.

Subset & proper subset

A⊆B if every element of A lies in B; A⊂B (proper) if additionally A≠B. Every set is a subset of itself and of the universal set.

Power set

The set of all subsets of A, written P(A). If n(A)=n then n(P(A))=2ⁿ, since each element is either in or out of any given subset.

Universal set

The fixed set U containing all objects under discussion; complements are taken relative to it. Every set in a problem is a subset of U.

Intervals as sets

Subsets of ℝ written with brackets: [a,b] includes endpoints, (a,b) excludes them. Their length is b−a and they contain infinitely many reals.

Union

A∪B is the set of elements in A, in B, or in both. It is commutative and associative, with A∪A = A, A∪∅ = A and A∪U = U.

Intersection

A∩B is the set of elements common to both A and B. If A∩B=∅ the sets are disjoint. Also A∩∅ = ∅ and A∩U = A.

Difference & complement

A−B keeps elements of A not in B. The complement A' = U−A is everything in U outside A, so A∪A' = U and A∩A' = ∅.

Venn diagrams

Diagrams showing sets as overlapping regions inside a rectangle for U. They make unions, intersections and De Morgan's laws visible at a glance.

De Morgan's laws

The complement of a union is the intersection of complements, and vice versa: (A∪B)' = A'∩B' and (A∩B)' = A'∪B'.

Inclusion–exclusion

For finite sets, n(A∪B) = n(A)+n(B)−n(A∩B); the overlap is subtracted once so that shared elements are not counted twice.