Set Test - 1

Number of elements in the power set of A = 2^x

Number of elements in the power set of B = 2^y

Given: 2^x = 2^y + 120

⇒ 2^x − 2^y = 120

⇒ 2^y (2^x−y − 1) = 2 × 2 × 2 × 3 × 5

⇒ 2^y (2^x−y − 1) = 2³ (2⁴ −1)

⇒ y = 3 and x − y = 4

∴ x = y + 4 = 3 + 4 = 7

Given set is A = {0, {1, Φ}, −1, 3}

I. It is observed that Φ∈{Φ}, but Φ∉A

∴ {Φ} ⊄ A

II. 1∈{1, Φ}, but 1∉A

∴ {1, Φ} ⊄ A

III. 0∈{0, −1}, −1∈{0, −1}

Also, 0∈A and −1∈A

∴{0, −1} ⊂ A

IV. Φ∈ {Φ, {1, Φ}, but Φ∉A

{Φ, {1, Φ}}⊄ A

Thus, three statements are incorrect.

It can be observed that 12 cannot be expressed in the form 2ⁿ (n ∈ N).

∴12∉A [ 12 = 2 × 2 × 3]

Thus, statement given in alternative (B) is incorrect.

The correct answer is B.

Why alternative A is wrong:

4² = (2 × 2) × (2 × 2) = 2⁴, where 4 ∈ N

∴ 4² ∈ A

Why alternative C is wrong:

For every value of natural number ‘n’, there is a different value of 2ⁿ.

Since N is an infinite set, A is an infinite set.

Why alternative D is wrong:

For every value of natural number ‘n’, there is a different value of 2ⁿ, which again is a natural number.

Thus, A is a subset of N.

Consider the statement given in alternative A

The word RHYTHM has no vowel. Therefore, the required set is empty.

Thus, the statement given in alternative A is correct.

Hence, the correct answer is option 1.

Solution:

The given set is {x: x is a prime number and divisible by 7}

The element of this set is {7}

∴ Cardinal number = 1

Thus, the cardinal number of the given set is 1.

The whole numbers less than 30, which are divisible by 5 are 0, 5, 10, 15, 20, 25

Thus, the required set is {0, 5, 10, 15, 20, 25}

The collection of all hard working students of a particular school is not well

defined because the criterion for determining a student as hard working may

vary from person to person.

Therefore, this collection is not a set.

Number of proper subsets of X = 2^a − 1

Number of proper subsets of Y = 2^b − 1

Given:

(2^a − 1) = (2^b − 1) + 224

⇒ 2^a = 2^b + 224

⇒2^a − 2^b = 224

⇒2^b(2^{a − b} − 1) = 2⁵(2³ − 1)

⇒ b = 5 and a − b = 3

∴ a = b + 3 = 5 + 3 = 8
 

The cardinality of set B is 11 i.e., m = 11. Thus, the number of proper subsets of set B is . 
 

Hence, statement A is correct.

The cardinality of set A is 4 i.e., m = 4. Thus, the number of subsets of set A is .

Hence, statement B is correct.

It can be observed that every element of set A is a member of set B and there exist elements in set B which are not the members of set A i.e., p, q, r, s ∈ B; however, t, u, v, w, x, y, z ∉ A. Thus, set A is a proper subset of set B.

Hence, statement D is correct.
 

Set A is a proper subset of set B.

Hence, statement C is incorrect.