Differentiation (Part-38) | S N De

Set Theory MCQs 24–44: Questions, Detailed Solutions and Important Concepts

This post contains English translations and detailed solutions of Questions 24 to 44 on Sets, Subsets, Power Sets, Cardinal Numbers, Empty Sets and Set Operations. Mathematical expressions are written using LaTeX notation.

Note: Questions 25 and 26 contain ambiguity/errors in the printed options. The mathematically correct conclusions are explained separately in their solutions.

Question No. 24

Question: For two sets \(A\) and \(B\), \(A-B=\varnothing\) if and only if which of the following is true?

• A. \(A\subseteq B\)
• B. \(B\subseteq A\)
• C. \(A\cap B=\varnothing\)
• D. None of these

Detailed Solution

The set difference \(A-B\) contains all elements that belong to \(A\) but do not belong to \(B\).

If \(A-B=\varnothing\), there is no element of \(A\) outside \(B\). Therefore, every element of \(A\) must belong to \(B\).

Hence, \[ A\subseteq B. \]

Conversely, if \(A\subseteq B\), then there is no element of \(A\) which is outside \(B\), so \(A-B=\varnothing\).

Answer: A. \(A\subseteq B\)

Question No. 25

Question: Given

\[ A=\{1,2,3\},\quad B=\{1,2\},\quad C=\{1,2,4\}, \] \[ D=\{3,4\},\quad E=\{4\}, \] which of the following statements is true?

• A. \(B\subset A\)
• B. \(D\not\subseteq E\)
• C. \(D\subset B\)
• D. \(\{E\}\subset A\)

Detailed Solution

We have \[ A=\{1,2,3\},\quad B=\{1,2\}. \] Every element of \(B\) belongs to \(A\), and \(B\neq A\). Therefore, \[ B\subset A. \] So statement A is true.

Also, \[ D=\{3,4\},\qquad E=\{4\}. \] Since \(3\notin E\), \(D\) is not a subset of \(E\). Thus statement B is also true.

Statement C is false because \(3,4\notin B\) completely; in particular \(4\notin B\).

For statement D, since \(E=\{4\}\), \[ \{E\}=\{\{4\}\}. \] But \(\{4\}\notin A\), so \(\{E\}\not\subset A\).

Important: As printed in the supplied image, both A and B are true. Therefore, Question 25 appears to contain an error/ambiguity in the original options.

Question No. 26

Question: \(A-B=B-A\) if and only if which condition holds?

• A. \(A\in B\)
• B. \(B\subset A\)
• C. \(A\subset B\)
• D. \(A\cap B=\varnothing\)

Detailed Solution

Notice that \(A-B\) and \(B-A\) are always disjoint.

If \[ A-B=B-A, \] then two equal sets are also disjoint. Therefore, both must be empty: \[ A-B=\varnothing \] and \[ B-A=\varnothing. \]

From \(A-B=\varnothing\), \[ A\subseteq B. \] From \(B-A=\varnothing\), \[ B\subseteq A. \] Hence, \[ A=B. \]

Therefore, the correct condition is \[ \boxed{A=B}. \]

Note: \(A=B\) is not present among the printed options. Also, \(A\cap B=\varnothing\) is not sufficient. For example, \(A=\{1\}\) and \(B=\{2\}\) give \(A-B=\{1\}\) and \(B-A=\{2\}\), which are not equal. Hence the printed question/options appear to contain an error.

Question No. 27

Question: Let

\[ A=\{x:x\text{ is a square}\}, \] \[ B=\{x:x\text{ is a quadrilateral}\}, \] \[ C=\{x:x\text{ is a rhombus}\}. \]

Which of the following statements is correct?

• A. \(A\subset C,\ C\subset A,\ A\subset B\)
• B. \(A\subset C,\ A\subset B,\ C\subset B\)
• C. \(A\subset B,\ B\subset C,\ A\subset C\)
• D. \(C\subset A,\ B\subset A,\ B\subset C\)

Detailed Solution

Every square is a rhombus because all four sides of a square are equal. Therefore, \[ A\subset C. \]

Every square is also a quadrilateral because it has four sides. Hence, \[ A\subset B. \]

Every rhombus is a quadrilateral. Therefore, \[ C\subset B. \]

Thus, \[ A\subset C,\quad A\subset B,\quad C\subset B. \]

Answer: B

Question No. 28

Question: Suppose \(X\) is the set of rational numbers. Which of the following is not a subset of \(X\)?

• A. The set of real numbers
• B. The set of natural numbers
• C. The set of negative integers
• D. The set of integers

Detailed Solution

Let \(X=\mathbb{Q}\), the set of rational numbers.

We know that \[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

Therefore, natural numbers, negative integers and integers are all subsets of the rational numbers.

However, every real number is not necessarily rational. For example, \[ \sqrt{2}\in\mathbb{R} \] but \[ \sqrt{2}\notin\mathbb{Q}. \]

Answer: A. The set of real numbers

Question No. 29

Question: If \(A\) is the set of even integers and \(B\) is the set of natural numbers, which statement is correct?

• A. \(A\subseteq B\)
• B. \(B\subseteq A\)
• C. \(A=B\)
• D. Both \(A\) and \(B\) are infinite sets

Detailed Solution

The set of even integers is \[ A=\{\ldots,-4,-2,0,2,4,6,\ldots\}. \] This set is infinite.

The set of natural numbers is also infinite: \[ B=\{1,2,3,4,\ldots\} \] or \(\{0,1,2,3,\ldots\}\), depending on convention.

\(A\) is not a subset of \(B\), because negative even integers belong to \(A\) but not to \(B\).

\(B\) is not a subset of \(A\), because odd natural numbers such as \(1,3,5\) do not belong to \(A\).

Therefore, both sets are infinite.

Answer: D. Both \(A\) and \(B\) are infinite sets

Question No. 30

Question: If \(B\) is the power set of \(A\), which statement is true?

• A. \(A\in B\)
• B. \(A\supset B\)
• C. \(A\subset B\)
• D. \(A=B\)

Detailed Solution

If \(B\) is the power set of \(A\), then \[ B=P(A). \]

The power set contains every subset of \(A\), including \(A\) itself. Therefore, \[ A\in P(A)=B. \]

Answer: A. \(A\in B\)

Question No. 31

Question: If \(A=\{3,4\}\), then the power set of \(A\) is:

• A. \(\{\{3\},\{4\}\}\)
• B. \(\{\{3\},\{4\},\{3,4\}\}\)
• C. \(\{\varnothing,\{3\},\{4\}\}\)
• D. \(\{\varnothing,\{3\},\{4\},\{3,4\}\}\)

Detailed Solution

The set \(A\) contains two elements: \[ |A|=2. \]

A set containing \(n\) elements has \(2^n\) subsets. Therefore, \[ |P(A)|=2^2=4. \]

The four subsets are \[ \varnothing,\quad \{3\},\quad \{4\},\quad \{3,4\}. \]

Hence, \[ P(A)=\{\varnothing,\{3\},\{4\},\{3,4\}\}. \]

Answer: D

Question No. 32

Question: If \(A=\{a,b,c\}\), which of the following statements is false?

• A. \(\varnothing\in A\)
• B. \(\varnothing\in P(A)\)
• C. \(\varnothing\subset A\)
• D. \(\varnothing\subset P(A)\)

Detailed Solution

The set \(A=\{a,b,c\}\) does not contain the empty set as an element. Therefore, \[ \varnothing\notin A. \] Hence statement A is false.

However, the empty set is a subset of every set. Therefore, \[ \varnothing\subset A. \]

Also, \(\varnothing\) is always an element of the power set: \[ \varnothing\in P(A). \]

Since \(P(A)\) is non-empty, \(\varnothing\) is also a proper subset of \(P(A)\).

Answer: A. \(\varnothing\in A\)

Question No. 33

Question: If \[ A=\{x:x\text{ is a prime number less than }4\}, \] then what is the cardinal number of \(P(P(A))\)?

• A. 8
• B. 16
• C. 32
• D. 64

Detailed Solution

The prime numbers less than \(4\) are \(2\) and \(3\). Thus, \[ A=\{2,3\}. \]

Therefore, \[ |A|=2. \]

The number of elements in the power set is \[ |P(A)|=2^2=4. \]

Therefore, \[ |P(P(A))|=2^4=16. \]

Answer: B. 16

Question No. 34

Question: Two finite sets contain \(m\) and \(n\) elements respectively. The number of subsets of the first set is 112 more than the number of subsets of the second set. Find \(m\) and \(n\), respectively.

• A. \(4,7\)
• B. \(7,4\)
• C. \(4,4\)
• D. \(7,7\)

Detailed Solution

If a set has \(m\) elements, its power set contains \(2^m\) elements. Similarly, a set with \(n\) elements has \(2^n\) subsets.

According to the question, \[ 2^m-2^n=112. \]

Check \(m=7,\ n=4\): \[ 2^7-2^4=128-16=112. \]

Hence, \[ m=7,\qquad n=4. \]

Answer: B. \(7,4\)

Question No. 35

Question: Which of the following sets is an empty set?

• A. \(A=\{x:x\in\mathbb Z\text{ and }3\le x\le5\}\)
• B. \(B=\{x:x\in\mathbb N\text{ and }8
• C. \(C=\{x:x\in\mathbb R,\ x^2+4=4\}\)
• D. \(D=\{x:x\in\mathbb N\text{ and }6x^2-7x+2=0\}\)

Detailed Solution

For A: \[ 3\le x\le5,\quad x\in\mathbb Z \] gives \[ A=\{3,4,5\}, \] so A is not empty.

For B: \[ 8

For C: \[ x^2+4=4 \] gives \[ x^2=0\Rightarrow x=0. \] Therefore C is not empty.

For D: \[ 6x^2-7x+2=0. \] Factoring, \[ (3x-2)(2x-1)=0. \] Hence, \[ x=\frac23\quad\text{or}\quad x=\frac12. \] Neither is a natural number.

Therefore, \[ D=\varnothing. \]

Answer: D

Question No. 36

Question: Which of the following is the set of odd positive integers divisible by \(2\)?

• A. \(\varnothing\)
• B. \(U\)
• C. \(\{0\}\)
• D. \(\{0\}\) or the printed set shown in the question

Detailed Solution

An odd positive integer cannot be divisible by \(2\).

For example, \[ 1,3,5,7,\ldots \] are odd, but none of them is divisible by \(2\).

Therefore, there is no odd positive integer divisible by \(2\). Hence the required set contains no elements: \[ \boxed{\varnothing}. \]

Answer: A. \(\varnothing\)

Question No. 37

Question: Which of the following sets is an empty set?

• A. \(A=\{x:x^2=16\text{ and }3x-5=7\}\)
• B. \(B=\{x:x\in\mathbb Z\text{ and }x^2-2=0\}\)
• C. \(C=\{x:x^2-3x+2=0\text{ and }x^2-4x+3=0\}\)
• D. \(D=\{x:x^2-16=0\text{ and }x\text{ is rational}\}\)

Detailed Solution

For A: \[ x^2=16\Rightarrow x=\pm4. \] Also, \[ 3x-5=7\Rightarrow3x=12\Rightarrow x=4. \] Therefore \(x=4\) satisfies both conditions, so A is non-empty.

For B: \[ x^2-2=0\Rightarrow x^2=2\Rightarrow x=\pm\sqrt2. \] But \(\sqrt2\) is irrational, so neither solution belongs to \(\mathbb Z\). Therefore, \[ B=\varnothing. \]

For C: \[ x^2-3x+2=0 \] gives \[ (x-1)(x-2)=0, \] so \(x=1,2\).

Also, \[ x^2-4x+3=0 \] gives \[ (x-1)(x-3)=0, \] so \(x=1,3\).

The common solution is \(x=1\), so C is non-empty.

For D: \[ x^2-16=0\Rightarrow x=\pm4. \] Both \(4\) and \(-4\) are rational, so D is non-empty.

Answer: B

Question No. 38

Question 38

Which pair of sets is disjoint?

• A. {1, 2, 3} and {x : x ∈ ℕ, 4 < x < 6}
• B. {a, e, l, o, u} and {c, d, e, f}
• C. {x : x is a multiple of 3, 5 ≤ x ≤ 30} and the set of even numbers
• D. None of these

Correct Answer: A. {1, 2, 3} and {x : x ∈ ℕ, 4 < x < 6}

Step-by-Step Solution

Key Concept: Disjoint Sets
Two sets A and B are called disjoint if they have no common elements. In set notation, their intersection must be an empty set:

A ∩ B = ∅

Evaluating Option A:

Given sets:

• First set: A = {1, 2, 3}
• Second set: B = {x : x ∈ ℕ, 4 < x < 6}
  Explanation: The natural numbers (ℕ) strictly between 4 and 6 contain only one number, which is 5. Thus, B = {5}.
• Intersection: A ∩ B = {1, 2, 3} ∩ {5} = ∅

Result: Since there are no shared elements, these sets are disjoint.

---

Evaluating Option B:

• First set: A = {a, e, l, o, u}
• Second set: B = {c, d, e, f}
• Intersection: Both sets contain the element 'e'.
  So, A ∩ B = {e} ≠ ∅

Result: Since they share element 'e', these sets are not disjoint.

---

Evaluating Option C:

• First set: Multiples of 3 between 5 and 30 = {6, 9, 12, 15, 18, 21, 24, 27, 30}
• Second set: Even numbers = {..., 2, 4, 6, 8, 10, 12, ...}
• Intersection: The numbers {6, 12, 18, 24, 30} belong to both sets.
  So, A ∩ B = {6, 12, 18, 24, 30} ≠ ∅

Result: Since even multiples of 3 exist in both sets, they are not disjoint.

Question No. 39

Question: Which of the following four sets is not an empty set?

• A. \(\varnothing\)
• B. \(\{x:x\text{ is the square of an odd number and }2\le x\le8\}\)
• C. \(\{x:x\text{ is a perfect number and }2 \lt x \lt 3 \}\)
• D. \(\{0\}\)

Detailed Solution

A is directly the empty set.

For B, the squares of odd integers are \[ 1,9,25,\ldots \] There is no such square between \(2\) and \(8\). Hence B is empty.

For C, there is no integer satisfying \[ 2 \lt x \lt 3 \]

D contains the element \(0\): \[ \{0\}\neq\varnothing. \]

Answer: D. \(\{0\}\)

Question No. 40

Question: If \(A\cap B=B\), then which statement is true?

• A. \(A\subseteq B\)
• B. \(A=B\)
• C. \(B\subseteq A\)
• D. None of these

Detailed Solution

The intersection \(A\cap B\) contains elements common to both \(A\) and \(B\).

If \[ A\cap B=B, \] it means every element of \(B\) is already contained in \(A\).

Therefore, \[ B\subseteq A. \]

Answer: C. \(B\subseteq A\)

Question No. 41

Question: If \[ A\cup B=A\cup C \] and \[ A\cap B=A\cap C, \] then which of the following is true?

• A. \(A=B\)
• B. \(B=C\)
• C. \(A=C\)
• D. None of these

Detailed Solution

We need to prove that \(B=C\).

Take any element \(x\in B\).

There are two possibilities.

Case 1: \(x\in A\). Then \[ x\in A\cap B. \] Since \[ A\cap B=A\cap C, \] we get \[ x\in C. \]

Case 2: \(x\notin A\). Since \(x\in B\), we have \[ x\in A\cup B. \] Given \[ A\cup B=A\cup C, \] we have \(x\in A\cup C\). Since \(x\notin A\), it follows that \(x\in C\).

Thus every element of \(B\) belongs to \(C\): \[ B\subseteq C. \]

By the same argument, \[ C\subseteq B. \] Hence, \[ B=C. \]

Answer: B. \(B=C\)

Question No. 42

Question: If \[ A=\{x:-1\le x\le6\} \] and \[ B=\{x:x>3\}, \] then find \(A\cap B\).

• A. \(\{x:3\le x\le6\}\)
• B. \(\{x:3\lt x\le6\}\)
• C. \(\{x:4\lt x\le6\}\)
• D. None of these

Detailed Solution

The first set gives \[ -1\le x\le6. \]

The second set gives \[ x>3. \]

For an element to belong to both sets, both conditions must hold: \[ x>3\quad\text{and}\quad x\le6. \]

Therefore, \[ A\cap B=\{x:3\lt x\le6\}\]

Answer: B

Question No. 43

Question: Which of the following is equal to \[ (A-B)\cup(B-A)? \]

• A. \((A\cup B)\cup(A-B)\)
• B. \((A\cup B)\cup(A\cap B)\)
• C. \((A\cup B)-(A\cap B)\)
• D. \((A-B)\cap(B-A)\)

Detailed Solution

The set \[ A-B \] contains elements that are in \(A\) but not in \(B\).

Similarly, \[ B-A \] contains elements that are in \(B\) but not in \(A\).

Their union therefore contains elements belonging to exactly one of the two sets. This is called the symmetric difference.

The symmetric difference can be written as \[ A\triangle B=(A\cup B)-(A\cap B). \]

Hence, \[ (A-B)\cup(B-A)=(A\cup B)-(A\cap B). \]

Answer: C

Question No. 44

Question: If \(A\) and \(B\) are two disjoint sets, then what is \[ A\cap(A\cap B)? \]

• A. \(A\)
• B. \(B\)
• C. \(\varnothing\)
• D. \(A\cap B\)

Detailed Solution

Since \(A\) and \(B\) are disjoint, \[ A\cap B=\varnothing. \]

Therefore, \[ A\cap(A\cap B) =A\cap\varnothing. \]

The intersection of any set with the empty set is the empty set: \[ A\cap\varnothing=\varnothing. \]

Hence, \[ \boxed{A\cap(A\cap B)=\varnothing}. \]

Answer: C. \(\varnothing\)

Important Concepts for Quick Revision

1. Set Difference

\[ A-B=\{x:x\in A\text{ and }x\notin B\}. \]

2. Important Subset Result

\[ A-B=\varnothing\iff A\subseteq B. \]

3. Equality of Set Differences

\[ A-B=B-A\iff A=B. \]

4. Empty Set

The empty set contains no elements: \[ \varnothing=\{\}. \]

The empty set is a subset of every set: \[ \varnothing\subseteq A. \]

5. Power Set

If \[ |A|=n, \] then \[ |P(A)|=2^n. \]

Also, \[ \varnothing\in P(A) \] and \[ A\in P(A). \]

6. Cardinal Number of a Double Power Set

If \[ |A|=n, \] then \[ |P(P(A))|=2^{2^n}. \]

7. Standard Number Sets

The usual inclusion relationship is \[ \mathbb N\subseteq\mathbb Z\subseteq\mathbb Q\subseteq\mathbb R. \]

8. Intersection

If \[ A\cap B=B, \] then \[ B\subseteq A. \]

9. Disjoint Sets

Two sets are disjoint if they have no common element: \[ A\cap B=\varnothing. \]

10. Symmetric Difference

The symmetric difference of \(A\) and \(B\) is \[ A\triangle B=(A-B)\cup(B-A). \]

An equivalent form is \[ A\triangle B=(A\cup B)-(A\cap B). \]

11. Useful Set Identities

\[ A\cap\varnothing=\varnothing \]

\[ A\cup\varnothing=A \]

\[ A\cap A=A \]

\[ A\cup A=A \]

\[ A\cap B=B\iff B\subseteq A. \]

\[ A\cup B=B\iff A\subseteq B. \]

Answer Summary

24 – A  |  25 – A and B*  |  26 – \(A=B\)*  |  27 – B  |  28 – A  |  29 – D  |  30 – A  |  31 – D  |  32 – A  |  33 – B  |  34 – B

35 – D  |  36 – A  |  37 – B  |  38 – A  |  39 – D  |  40 – C  |  41 – B  |  42 – B  |  43 – C  |  44 – C

*Note: Questions 25 and 26 have apparent inconsistencies in the supplied printed options. The mathematically correct conclusions are explained in their respective solutions.