Set Theory and types of Sets MCQ Quiz in मल्याळम - Objective Question with Answer for Set Theory and types of Sets - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept Used:

P(A) represents the Powerset of A i.e., the set of all subsets of A.

|P(A)| = 2^|A|

A - B = {x| x ∈ A and x ∉ B}

Calculation:

|A| = 50, |A ∩ B| = 45 and |B| = 48

⇒ A and B have 45 elements in common.

Hence, A - B must have only 5 elements i.e., |A - B| = 5

Hence, |P(A − B)| = 2^{|(A − B)|} = 2⁵

CONCEPT:

Symmetric Difference of Sets:

Let A and B be two sets. The symmetric difference of sets A and B is the set (A - B) ∪ (B - A) and is denoted as A Δ B i.e A Δ B = (A - B) ∪ (B – A).

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

CALCULATION:

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

First let's find out A - B and B - A

⇒ A - B = {1, 5, 7} and B - A = {4, 6}

As we know that, A Δ B = (A - B) ∪ (B – A)

⇒ A Δ B = {1, 5, 7} ∪ {4, 6} = {1, 4, 5, 6, 7}

As we can see that, the number if elements present in A Δ B = 5 i.e n(A Δ B) = 5

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

So, the number of proper subsets of A Δ B = 2⁵ - 1 = 31

Hence, the correct option is 3.

Concept use:

A - B = The element which are present in A but not in B

Calculations:

If set A = {1, 3, 5, 7}, set B = {1, 4, 7}, 

The Value of A - B = {3, 5} (The element which are present in A but not in B)

Concept:

Subset: Let A and B be two sets. If every element of A is present in set B then A is called subset of B and it is denoted as A ⊆ B.

Equal Sets: Any two sets are said to be equal sets if and only if they are equivalent and as well as their elements are same.

Calculation:

Given: A = {1, 2, 3, 4} and B = {x ∈ N : x ≤ 5}

The set B can be re-written as B = {1, 2, 3, 4, 5}

As we know that, if A and B are two sets such that every element of A is present in set B but there is one element in B which is not present in A then A is a proper subset of B and is denoted as A ⊂ B

As we can see that, all the elements of set A are present in B but we have 5 ∈ B but 5 ∉ A ⇒ A ⊂ B

Hence, option 1 is the correct answer.

Concept:

Power set: Let A be a non-empty set. Then, $P\left(A\right)=\left\{B|B\subseteq A\right\}$ is called as the power set of A.

For any non-empty set A with n(A) = x. Then the total no. of subsets of set A is given by: 2^x.

i.e n(P(A)) = 2^x.

Calculation:

Given: A = {λ, {λ, μ}}

Subsets of A are: {λ}, {{λ, μ}}, {λ, {λ, μ}}, φ

Concept Used:

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

Calculation:
let n(M) be the number of teachers who can teach mathematics and n(P) be the number of teachers who can teach physics.
Given: n(M ∪ P) = 20, n(M ∩ P) = 4, n(M) = 12
We have to calculate no of teachers who can only teach physics  i.e., n(P) - n(P ∩ M) 

Now, n(P ∪ M) = n(P) + n(M) - n(P ∩ M) 

⇒  20 = n(P) + 12 - 4 

⇒ n(P) = 20 - 8 = 12

But this represents the total number of teachers teaching Physics, including those who teach both subjects. To find the number of teachers who teach only Physics, we subtract those who teach both subjects:
⇒ n(P) - n(P∩M) = 12 - 4 = 8

Hence, the number of teachers who can only teach physics is 8.

Concept:

• A ⋃ B denotes the elements which are in set A or set B.
• A ∩ B denotes the elements which are in both set A and set B.
• A-B denotes the elements which are only in set A and not in B.
• B-A denotes the elements which are only in set B and not in set A.

Calculation:

As it can be seen from the Venn diagram the purple area is (B - A) and the orange area is (A - B).

The green area is (A ∩ B) and all three of those areas together are (A ⋃ B).

It can be seen that (A - B) ⋃ (B - A) is the orange and the purple areas together.

This can also be found in the difference between (A ⋃ B) and (A ∩ B).

Hence, (A - B) ⋃ (B - A) = (A ⋃ B) - (A ∩ B)

Concept:

• Null set is the set that does not contain anything.
• In mathematical sets, the null set, also called the empty set,
• It is symbolized or { }.

Calculation:

Given:

A = {x : x is a multiple of 3}

∴ A = {3, 6, 9, 12, 15, 18, 24…}

B = {x : x is a multiple of 4}

∴ B = {4, 8, 12, 16, 20, 24, 28, 32 ...}

C = {x : x is a multiple of 12}

∴ C = {12, 24, 36, 48, 60, 72, 84, 96 ...}

Now,

A∩B = {12, 24 ...} = C

∴ (A∩B) \ C = { } = Null set

Concept:

Let A be set, then the set of all the possible subsets of A is called the power set of A is called the power set of A and is denoted by P(A) = {X : X ⊆ A}

Calculation:

Given: A = {1, a, 3}

⇒ the subsets of A are: ϕ, {1}, {a}, {3}, {1, a}, {1, 3}, {a, 3}, A

As we know that, 

The power set of A  is given by P(A) = {X : X ⊆ A}

⇒ P(A) = {ϕ, {1}, {a}, {3}, {1, a}, {1, 3}, {a, 3}, A}

Hence, option 1 is correct.

Concept:

If A and B are two sets, then their difference is given by A - B or B - A. 

If A = {2, 3, 4} and B = {4, 5, 6} 

A - B means elements of A that are not the elements of B i.e,

in the above example A - B = {2, 3} 

In general, B - A = {x : x ∈ B, and x ∉ A} 

Proper subset = 2ⁿ - 1,

n is the number of elements

Calculation:

Given:

Number of elements in X = 6

Number of eleelementsent in Y = 5

Number of elements in Z = 4

All elements are different.

Since, X, Y,  have distnict elements. 

Therefore,

n(X - Y) = 6, n(Z) = 4

⇒ S = (X - Y) ∪ Z

⇒ S = 6 + 4 = 10

Hence,

Number of proper subsets 

N = 2¹⁰ - 1

⇒ N = 1024 - 1

⇒ N = 1023

S has 1023 proper subsets.