Set Theory and types of Sets MCQ Quiz - Objective Question with Answer for Set Theory and types of Sets - Download Free PDF

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Latest Set Theory and types of Sets MCQ Objective Questions

Set Theory and types of Sets Question 1:

If a set has 5 elements, then the power set of that set has ______ elements.

  1. 25
  2. 32
  3. 10
  4. None of these

Answer (Detailed Solution Below)

Option 2 : 32

Set Theory and types of Sets Question 1 Detailed Solution

Given:

Number of elements in the set = 5

Formula Used:

Number of elements in a power set = 2n, where n is the number of elements in the set.

Calculation:

n = 5

⇒ Number of elements in the power set = 25

⇒ Number of elements in the power set = 2 × 2 × 2 × 2 × 2

⇒ Number of elements in the power set = 32

The power set of a set with 5 elements has 32 elements.

Set Theory and types of Sets Question 2:

Which of the following is a countable set?

  1. Set of all irrational numbers
  2. Set of all natural numbers
  3. All values in the closed interval [0, 1]
  4. Set of all complex numbers

Answer (Detailed Solution Below)

Option 2 : Set of all natural numbers

Set Theory and types of Sets Question 2 Detailed Solution

The correct answer is Set of all natural numbers.

Key Points

  • A set is considered countable if it has the same cardinality as the set of natural numbers or if it is finite.
    • The set of all natural numbers (denoted as N = {1, 2, 3, ...}) is an example of a countable infinite set because each element can be uniquely paired with a natural number.
    • Countable sets can either be finite or have a one-to-one correspondence with the natural numbers.
  • Other sets in the options are not countable:
    • The set of all irrational numbers is uncountable because it is a subset of the real numbers with infinite cardinality that cannot be paired with natural numbers.
    • The set of all values in the closed interval [0, 1] is uncountable because it contains all real numbers between 0 and 1, which form a continuum.
    • The set of all complex numbers is uncountable because it includes all possible combinations of real and imaginary numbers.

Additional Information

  • In mathematics, the concept of countability is essential in set theory and helps classify the size of infinite sets.
  • The set of integers and the set of rational numbers are also countable because they can be arranged in a sequence that corresponds to natural numbers.
  • The concept of countability was formalized by mathematician Georg Cantor, who developed set theory.
  • The study of uncountable sets, such as the real numbers, led to the development of advanced mathematical fields like topology and measure theory.

Set Theory and types of Sets Question 3:

If Set A = {1, 2, 3} and Set B = {1), which of the following is true?

  1. A - B={2,3}
  2. (A intersect B) = {1, 2, 3}
  3. A x B={1, 2, 3}
  4. A delta B = {1}

Answer (Detailed Solution Below)

Option 1 : A - B={2,3}

Set Theory and types of Sets Question 3 Detailed Solution

The correct answer is Option 1) A - B = {2, 3}.

Key Points

  • Set A = {1, 2, 3}
  • Set B = {1}
  • A - B means elements in A that are not in B.
  • So: A - B = {2, 3}

Additional Information

  • Option 2 – Incorrect: A ∩ B = {1}, not {1, 2, 3}.
  • Option 3 – Incorrect: A × B = {(1,1), (2,1), (3,1)} — Cartesian product, not a set of numbers.
  • Option 4 – Incorrect: A Δ B (symmetric difference) = (A ∪ B) - (A ∩ B) = {2, 3}.
  • Summary:
    • Set difference (A - B): Elements in A but not in B
    • Intersection (A ∩ B): Elements common to both
    • Union (A ∪ B): All elements from both sets
    • Symmetric difference (A Δ B): Elements in A or B but not both
    • Cartesian product (A × B): All ordered pairs (a, b) such that a ∈ A, b ∈ B

Set Theory and types of Sets Question 4:

What is the count of odd-sized subsets for a set of size 5?

  1. 17
  2. 15
  3. 14
  4. 16

Answer (Detailed Solution Below)

Option 4 : 16

Set Theory and types of Sets Question 4 Detailed Solution

The correct answer is Option 4 (16).

Key Points

  • A set of size 5 has 25 = 32 subsets in total.
  • Out of these 32 subsets, half of them (16 subsets) will have an odd size, and the other half will have an even size. This is because for any set, the total subsets are evenly distributed between odd-sized subsets and even-sized subsets.
  • Odd-sized subsets are subsets with sizes 1, 3, or 5 (since these are the odd numbers ≤ 5).
  • The counts of subsets for each odd size can be calculated using the binomial coefficient formula C(n, k) = n! / [k!(n-k)!], where n is the size of the set, and k is the size of the subset:
    • Subsets of size 1: C(5, 1) = 5
    • Subsets of size 3: C(5, 3) = 10
    • Subsets of size 5: C(5, 5) = 1
  • Adding these counts together: 5 + 10 + 1 = 16.

Additional Information

  • The binomial coefficient C(n, k) represents the number of ways to choose k items from a set of n items.
  • The formula for calculating subsets ensures that all combinations are accounted for, including both odd and even sizes.
  • For sets of even size, the number of odd-sized subsets always equals the number of even-sized subsets.
  • This property is a consequence of the symmetry of the binomial coefficients in Pascal's Triangle.
  • For sets of odd size, there will always be one more odd-sized subset than even-sized subsets.

Set Theory and types of Sets Question 5:

Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non empty subsets of S that have the sum of all elements a multiple of 3, is ____ . 

Answer (Detailed Solution Below) 43

Set Theory and types of Sets Question 5 Detailed Solution

Calculation: 

Elements of the type 3k = 3 

Elements of the type 3k + 1 = 1, 7, 9

Elements of the type 3k + 2 = 2, 5, 11

Subsets containing one element S1 = 1

Subsets containing two elements

⇒ S2 = 3C1 × 3C1 = 9

Subsets containing three elements

⇒ S3 = 3C1 × 3C1 + 1 + 1 = 11

Subsets containing four elements

⇒ S4 = 3C3 + 3C3 + 3C2 × 3C2 = 11

Subsets containing five elements

⇒ S5 = 3C2 × 3C2 × 1 = 9

Subsets containing six elements S6 = 1

Subsets containing seven elements S7 = 1 

⇒ sum = 43 

Hence, the correct answer is 43. 

Top Set Theory and types of Sets MCQ Objective Questions

If A = {1, 3, 4} and B = {x : x ∈ R and x2 - 7x + 12 = 0} then which of the following is true ?

  1. A = B
  2. A ⊂ B
  3. B ⊂ A
  4. A is equivalent to B

Answer (Detailed Solution Below)

Option 3 : B ⊂ A

Set Theory and types of Sets Question 6 Detailed Solution

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CONCEPT:

Let A and B be two sets then A is said to be proper subset of B, if A is a subset of B and A is not equal to B. It is denoted as A ⊂ B.

CALCULATION:

Given: A = {1, 3, 4} and B = {x : x ∈ R and x2 - 7x + 12 = 0}

First let's find the roaster form of set B

In order to do so we need to find the roots of the equation x2 - 7x + 12 = 0

⇒ x2 - 3x - 4x + 12 = 0

⇒ x(x - 3) - 4(x - 3) = 0

⇒ (x - 4) × (x - 3) = 0

⇒ x = 3, 4

⇒ B = {3, 4}

As we can clearly see that, all the elements of B are there in set A but A ≠ B i.e B ⊂ A

Hence, the correct option is 3.

If A = {1, 2, 5, 7} and B = {2, 4, 6} then find the number of proper subsets of A U B ?

  1. 127
  2. 64
  3. 63
  4. 31

Answer (Detailed Solution Below)

Option 3 : 63

Set Theory and types of Sets Question 7 Detailed Solution

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CONCEPT:

Union:

Let A and B be two sets. The union of A and B is the set of all those elements which belong to either A or B or both A and B i.e A ∪ B = {x : x ∈ A or x ∈ B}

Note: If A is a non-empty set such that n(A) = m then numbers of proper subsets of A is given by 2m - 1.

CALCULATION:

Given: A = {1, 2, 5, 7} and B = {2, 4, 6}

As we know that,  A ∪ B = {x : x ∈ A or x ∈ B}.

⇒ A ∪ B = {1, 2, 4, 5, 6, 7}

As we can see that, the number of elements present in A U B = 6 i.e n(A U B) = 6

As we know that, if A is a non-empty set such that n(A) = m then numbers of proper subsets of A is given by 2m - 1.

So, the number of proper subsets of A Δ B = 26 - 1 = 63

Hence, the correct option is 3.

Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing exactly two elements is

  1. 20
  2. 40
  3. 45
  4. 90

Answer (Detailed Solution Below)

Option 3 : 45

Set Theory and types of Sets Question 8 Detailed Solution

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Concept:

Combination: Selecting r objects from given n objects.

  • The number of selections of r objects from the given n objects is denoted by 
  •  


Note: Use combinations if a problem calls for the number of ways of selecting objects.

Calculation:

Number of elements in A = 10

Number of subsets of A containing exactly two elements = Number of ways we can select 2 elements from 10 elements

⇒ Number of ways we can select 2 elements from 10 elements = 10C2 = 45

∴ Number of subsets of A containing exactly two elements = 45

If A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9} then find the number of proper subsets of A ∩  B ?

  1. 16
  2. 15
  3. 32
  4. 31

Answer (Detailed Solution Below)

Option 2 : 15

Set Theory and types of Sets Question 9 Detailed Solution

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CONCEPT:

Intersection:

Let A and B be two sets. The intersection of A and B is the set of all those elements which are present in both sets A and B.

The intersection of A and B is denoted by A ∩ B i.e A ∩ B = {x : x ∈ A and x ∈ B}

Note: If A is a non-empty set such that n(A) = m then numbers of proper subsets of A is given by 2m - 1.

CALCULATION:

Given: A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9}

As we know that, A ∩ B = {x : x ∈ A and x ∈ B}

⇒ A ∩ B = {2, 4, 7, 9}

As we can see that,

The number of elements present in A ∩ B = 4

i.e n(A ∩ B) = 4

As we know that;

If A is a non-empty set such that n(A) = m then

The numbers of proper subsets of A are given by 2m - 1.

So, The number of proper subsets of A ∩  B = 24 - 1 = 15

Hence, the correct option is 2

If A = {1, 3, 5} then find the cardinality of the power set of A ?

  1. 4
  2. 6
  3. 9
  4. 8

Answer (Detailed Solution Below)

Option 4 : 8

Set Theory and types of Sets Question 10 Detailed Solution

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CONCEPT:

Power Set:

Let A be a set, then the set of all the possible subsets of A is called the power set of A and is denoted by P(A).

Note: If A is a finite set with m elements. Then the number of elements (cardinality) of the power set of A is given by: n (P(A)) = 2m.

CALCULATION:

Given: A = {1, 3, 5}

Here, we have to find the cardinality of the power set of A i.e n (P(A))

As we know that if A is a finite set with m elements. Then the number of elements (cardinality) of the power set of A is given by: n (P(A)) = 2m.

Here, we can see that, the given A has 3 elements i.e n(A) = 3

So, the cardinality of the given set is n(P(A)) = 23 = 8

Hence, the correct option is 4.

If B = {x : x ∈ N such that x2 + 11x + 30 = 0} then B is a/an ?

  1. Empty set
  2. Finite set
  3. Infinite set
  4. None of these

Answer (Detailed Solution Below)

Option 1 : Empty set

Set Theory and types of Sets Question 11 Detailed Solution

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CONCEPT:

  • A set which contains finite number of elements is called a finite set.
  • A set which has infinite number of elements is called an infinite set
  • A set which does not contain any element is called an empty set.


CALCULATION:

Given: B = {x : x ∈ N such that x2 + 11x + 30 = 0}

First let's solve the quadratic equation x2 + 11x + 30 = 0

⇒ x2 + 5x + 6x + 30 = 0

⇒ x(x + 5) + 6(x + 5) = 0

⇒ (x + 6) (x + 5) = 0

⇒ x = - 5 or - 6

According to the definition of the given set x is a natural number but we know that neither x = - 5 nor x = - 6 is a natural number

So, the given set is an empty set i.e B = ϕ

Hence, the correct option is 1.

If A = {x, y, z}, then the number of subsets in powerset of A is

  1. 6
  2. 8
  3. 7
  4. 9

Answer (Detailed Solution Below)

Option 2 : 8

Set Theory and types of Sets Question 12 Detailed Solution

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Concept:

  • The power set (or powerset) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set.

Calculations:

Given, A = {x, y, z}. 

The power set (or powerset) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set.

Powerset of A = {ϕ,{x}, {y}, {z}, {x, y}, {y, z}, {x, z},{x, y, z}}.

Hence, the number of subsets in the powerset of A is 8.

In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) which of the following is correct:

  1. There exist at least two natural numbers which are prime to each other
  2. exist at least three natural number which are prime to each other
  3. There exist no consecutive natural number
  4. There exist more than two natural numbers which are prime to each other

Answer (Detailed Solution Below)

Option 1 : There exist at least two natural numbers which are prime to each other

Set Theory and types of Sets Question 13 Detailed Solution

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Concept:

The Pigeonhole Principle: Let there be n boxes and (n + 1) objects. Then, under any assignment of objects to the boxes, there will always be a box with more than one object in it. This can be reworded as, if m pigeons occupy n pigeonholes, where m > n, then there is at least one pigeonhole with two or more pigeons in it. 

Calculation:        

We divide the set into n classes {1, 2}, {3, 4},......{2n - 1, 2n}.

By the pigeonhole principle, given n +1 elements at least two of them will be in the same case {2k - 1, 2k} (1 ≤ k ≤ n). But 2k - 1 and 2k are relatively prime because their difference is 1. 

Consider the following statements:

1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.

2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets.

Which of the above statements is/are correct?

  1. 1 only
  2. 2 only
  3. Both 1 and 2
  4. Neither 1 nor 2

Answer (Detailed Solution Below)

Option 3 : Both 1 and 2

Set Theory and types of Sets Question 14 Detailed Solution

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Concept:

Equivalent set: Two sets A and B are said to be equivalent if they have the same number of elements (cardinality), regardless of what the elements are. , i.e n(A) = n(B).

Equal set: Two sets A and B are said to be equal if they have exactly the same elements. The repetition of elements does not matter in sets because a set is defined as a collection of distinct objects, so repeated elements are generally ignored.

If A = B, then n(A) = n(B) and for any x ∈ A, x ∈ B too.

Calculation:

1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets. → True

Set A has three elements: 1, 3, and 5. Set B also has three elements: 2, 4, and 7. Since both sets have three elements, they are indeed equivalent sets.

2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets. True

Set A contains the elements 1, 5, and 9. Set B also contains the elements 1, 5, and 9 (despite some elements being repeated, duplicates are not considered in set theory). Since both sets have the same elements, they are equal.

Therefore, both statements are correct:

Statement 1 is correct: A and B are equivalent sets.
Statement 2 is correct: A and B are equal sets.

Hence, The correct answer is "Both 1 and 2".

Source:- https://ncert.nic.in/textbook/pdf/kemh101.pdf, Page No. 7-8.

The power set of {0, 1, 2, …, 9} is?

  1. 512
  2. 256
  3. 1024
  4. 500

Answer (Detailed Solution Below)

Option 3 : 1024

Set Theory and types of Sets Question 15 Detailed Solution

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Concept:

The number of elements in the power set of any set A is 2n where n is the number of elements of the set A.

Calculation:

Let A = {0, 1, 2, …, 9}

Number of elements in set A = 10

∴ The number of element in the power set P(A) = 210 = 1024

Hence, option (3) is correct.

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