Sets: Exercise 1.2 – Guided Solutions
Work through every question with crystal‑clear steps, stress–strain visuals and exam‑oriented hints – all in a sleek deep‑space layout.
Q1
Which of the following are examples of the null set?
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) \( \{x : x \text{ is a natural number},\; x < 5 \text{ and } x> 7\} \)
(iv) \( \{y : y \text{ is a point common to any two parallel lines}\} \)
Concept Theory
A null set (or empty set) is a set that contains no elements. It is denoted by
\( \varnothing \quad \text{or} \quad \{\} \)
A set becomes a null set when the condition used to define the set cannot be satisfied by any element.
Illustration of a Null Set (No Elements Inside)
Solution Roadmap
To determine whether a given collection is a null set:
- Understand the condition defining the set.
- Check whether any element satisfies the condition.
- If no element satisfies the condition → Null Set.
- If at least one element satisfies the condition → Not a Null Set.
Solution
(i) Set of odd natural numbers divisible by 2
Odd numbers are not divisible by 2 because divisibility by 2
is the defining property of even numbers.
\( \Rightarrow \) No such number exists.
Therefore, this set has no element.
\( \Rightarrow \) It is a null set.
(ii) Set of even prime numbers
The number \(2\) is both even and prime.
Set = \( \{2\} \)
Since the set contains one element, it is not a null set.
(iii) \( \{x : x \text{ is a natural number}, x < 5 \text{ and } x> 7\} \)
A number cannot simultaneously be
\( x < 5 \) and \( x> 7 \)
Therefore, the condition is impossible to satisfy.
\( \Rightarrow \) The set has no elements.
Hence it is a null set.
(iv) \( \{y : y \text{ is a point common to any two parallel lines}\} \)
Two parallel lines in a plane never intersect.
Therefore, there is no common point.
\( \Rightarrow \) This set is also a null set.
The examples of the null set are:
\( (i), (iii), (iv) \)
Why This Question is Important
- Tests the basic understanding of the null set concept.
- Strengthens logical reasoning with set-builder conditions.
- Builds foundation for later topics like subsets, unions, intersections, and Venn diagrams.
- Common conceptual question type in JEE, NDA, and other entrance exams.
Q2
Which of the following sets are finite or infinite:
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . . 99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Concept Theory
In set theory, sets are classified according to the number of elements they contain.
- A finite set contains a fixed and countable number of elements.
- An infinite set contains elements that continue indefinitely and cannot be completely counted.
Solution Roadmap
To determine whether a set is finite or infinite:
- Observe the pattern of elements.
- Check whether the list stops at a specific number.
- If there is a last element → Finite.
- If the sequence continues forever → Infinite.
Solution
(i) The set of months of a year
A year contains exactly 12 months:
January, February, March, … , December
Therefore the set has a fixed number of elements.
\( \Rightarrow \) It is a finite set.
(ii) \( \{1,2,3,\ldots\} \)
This represents the set of natural numbers beginning from 1.
\(1,2,3,4,5,\ldots\)
There is no largest natural number.
\( \Rightarrow \) The set continues without end.
Hence, it is an infinite set.
(iii) \( \{1,2,3,\ldots,99,100\} \)
The elements start from 1 and end at 100.
Therefore the set contains exactly
\(100\) elements.
\( \Rightarrow \) This is a finite set.
(iv) The set of positive integers greater than 100
The elements are
\(101,102,103,104,\ldots\)
These numbers continue indefinitely.
\( \Rightarrow \) Hence it is an infinite set.
(v) The set of prime numbers less than 99
Prime numbers less than 99 include
\(2,3,5,7,11,\ldots,97\)
Only a limited number of primes exist below 99.
\( \Rightarrow \) Therefore this set is finite.
- (i) Finite
- (ii) Infinite
- (iii) Finite
- (iv) Infinite
- (v) Finite
Why This Question is Important
- Builds the foundation for understanding cardinality of sets.
- Helps students recognize patterns in number sequences.
- Important for topics like countability, subsets, and infinite sets.
- Frequently tested conceptually in JEE, NDA, Olympiads, and other entrance exams.
Q3
State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiples of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
Concept Theory
Sets are classified according to the number of elements they contain.
- A finite set contains a limited number of elements that can be counted.
- An infinite set contains elements that continue without end.
Solution Roadmap
- Identify the rule defining the set.
- Determine whether the number of elements is limited.
- If elements can be endlessly generated → Infinite.
- If elements are limited → Finite.
Solution
(i) The set of lines parallel to the x-axis
Any line parallel to the x-axis has the equation
\( y = c \)
where \( c \) is a real number.
Since infinitely many real numbers exist, infinitely many such lines can be drawn.
\( \Rightarrow \) The set is infinite.
(ii) The set of letters in the English alphabet
The English alphabet contains exactly
\( 26 \) letters.
Therefore the number of elements is fixed.
\( \Rightarrow \) This set is finite.
(iii) The set of numbers which are multiples of 5
Multiples of 5 are
\( 5,10,15,20,25,\ldots \)
This sequence continues indefinitely.
\( \Rightarrow \) The set is infinite.
(iv) The set of animals living on the earth
At any particular time, the number of animals on Earth is limited and countable.
Therefore the number of elements is bounded.
\( \Rightarrow \) This set is finite.
(v) The set of circles passing through the origin (0,0)
A circle in coordinate geometry is defined by its centre and radius.
By choosing infinitely many centres, infinitely many circles can pass through the origin.
\( \Rightarrow \) Therefore the set is infinite.
- (i) Infinite
- (ii) Finite
- (iii) Infinite
- (iv) Finite
- (v) Infinite
Why This Question is Important
- Develops understanding of finite vs infinite sets.
- Connects set theory with coordinate geometry.
- Strengthens reasoning about mathematical sequences and geometric objects.
- Concept often tested in JEE, Olympiads, and foundation mathematics exams.
Q4
In the following, state whether \(A = B\) or not:
\(
\begin{array}{l}
(i)\;A = \{ a, b, c, d \} \quad B = \{ d, c, b, a \}\\
(ii)\;A = \{ 4, 8, 12, 16 \} \quad B = \{ 8, 4, 16, 18 \}\\
(iii)\;A = \{2, 4, 6, 8, 10\} \quad B = \{ x : x\text{ is positive even integer and }x \le 10\}\\
(iv)\;A = \{ x : x\text{ is a multiple of }10\} \quad B = \{ 10, 15, 20, 25, 30, \ldots \}
\end{array}
\)
Concept Theory
Two sets \(A\) and \(B\) are said to be equal if they contain exactly the same elements.
\( A = B \iff \text{every element of } A \in B \text{ and every element of } B \in A \)
Equal sets contain identical elements
Solution Roadmap
- List the elements of both sets.
- Check whether each element of \(A\) is in \(B\).
- Check whether each element of \(B\) is in \(A\).
- If both conditions hold → \(A = B\).
Solution
(i) \(A = \{a,b,c,d\}\), \(B = \{d,c,b,a\}\)
Both sets contain the same four elements:
\(a,b,c,d\)
The order of elements is different, but sets ignore order.
\( \Rightarrow A = B \)
(ii) \(A = \{4,8,12,16\}\) \(B = \{8,4,16,18\}\)
Compare elements:
- 12 is in \(A\) but not in \(B\)
- 18 is in \(B\) but not in \(A\)
Since both sets do not contain exactly the same elements:
\( \Rightarrow A \ne B \)
(iii)
\(A = \{2,4,6,8,10\}\)
\(B = \{x : x \text{ is positive even integer and } x \le 10\}\)
Positive even integers less than or equal to 10 are:
\(2,4,6,8,10\)
Therefore both sets contain identical elements.
\( \Rightarrow A = B \)
(iv)
\(A = \{x : x \text{ is a multiple of }10\}\)
Elements of \(A\) are:
\(10,20,30,40,\ldots\)
But set \(B\) is
\(10,15,20,25,30,\ldots\)
The numbers \(15\) and \(25\) are not multiples of 10.
Therefore the two sets are different.
\( \Rightarrow A \ne B \)
- (i) \(A = B\)
- (ii) \(A \ne B\)
- (iii) \(A = B\)
- (iv) \(A \ne B\)
Why This Question is Important
- Builds understanding of set equality.
- Clarifies that order does not matter in sets.
- Introduces connection between roster form and set-builder form.
- Important foundation for later topics such as subset, union, and intersection.
Q5
Q5.
Are the following pairs of sets equal? Give reasons.
\(
\begin{array}{l}
(i)\; A = \{2,3\} \quad B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}\\
(ii)\; A = \{x : x \text{ is a letter in the word FOLLOW}\} \quad
B = \{y : y \text{ is a letter in the word WOLF}\}
\end{array}
\)
Concept Theory
Two sets are said to be equal if they contain exactly the same elements, regardless of the order in which the elements are written.
\( A = B \iff \text{each element of } A \in B \text{ and each element of } B \in A \)
Solution Roadmap
- Identify the elements of each set.
- If necessary, solve equations or simplify the definition.
- Compare the resulting elements of both sets.
- If all elements match → Sets are equal.
Solution
(i)
\(A = \{2,3\}\)
\(B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}\)
Solve the quadratic equation:
\(x^2 + 5x + 6 = 0\)
\( (x+2)(x+3)=0 \)
\(x=-2 \quad \text{or} \quad x=-3\)
Therefore,
\(B=\{-2,-3\}\)
Compare the sets:
- \(A = \{2,3\}\)
- \(B = \{-2,-3\}\)
The elements are different.
\( \Rightarrow A \ne B \)
(ii)
\(A = \{x : x \text{ is a letter in the word FOLLOW}\}\)
The word FOLLOW contains the letters:
F, O, L, L, O, W
In sets, repeated elements are written only once.
\(A = \{F,O,L,W\}\)
Now consider
\(B = \{y : y \text{ is a letter in the word WOLF}\}\)
Letters in WOLF are:
W, O, L, F
\(B = \{W,O,L,F\}\)
Both sets contain exactly the same letters.
\( \Rightarrow A = B \)
- (i) \(A \ne B\)
- (ii) \(A = B\)
Why This Question is Important
- Shows how sets can be represented in different forms (equations, words, or lists).
- Demonstrates that repeated elements are ignored in sets.
- Combines algebra (solving equations) with set theory.
- Strengthens conceptual understanding of set equality.
Q6
From the sets given below, select equal sets :
A = {2, 4, 8, 12}
B = {1, 2, 3, 4}
C = {4, 8, 12, 14}
D = {3, 1, 4, 2}
E = {–1, 1}
F = {0, a}
G = {1, –1}
H = {0, 1}
Concept Theory
Two sets are said to be equal if they contain exactly the same elements.
\( A = B \iff \text{both sets have identical elements} \)
Solution Roadmap
- List elements of each set.
- Compare the elements of different sets.
- If two sets contain identical elements → equal sets.
- If even one element differs → sets are not equal.
Solution
The given sets are:
\(A = \{2,4,8,12\}\)
\(B = \{1,2,3,4\}\)
\(C = \{4,8,12,14\}\)
\(D = \{3,1,4,2\}\)
\(E = \{-1,1\}\)
\(F = \{0,a\}\)
\(G = \{1,-1\}\)
\(H = \{0,1\}\)
Compare the sets:
\(B = \{1,2,3,4\}\)
\(D = \{3,1,4,2\}\)
Both sets contain the same elements \(1,2,3,4\), only the order is different.
\( \Rightarrow B = D \)
Next compare:
\(E = \{-1,1\}\)
\(G = \{1,-1\}\)
Both sets contain the same elements \(1\) and \(-1\).
\( \Rightarrow E = G \)
All other sets contain at least one different element, therefore they are not equal.
- \(B = D\)
- \(E = G\)
Why This Question is Important
- Strengthens understanding of set equality.
- Reinforces that order does not matter in sets.
- Helps students compare sets logically.
- Forms the basis for later topics like subset, union, and intersection.
Interactive Set Comparison Visualizer
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Set A
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Set B
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1️⃣ Set Builder → Roster Converter
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3️⃣ 3-Set Venn Diagram Engine
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