Test: Set Notations - JEE MCQ

Calculation:

The set X is given by

{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98}.

We want to find the maximum size of a subset S of X such that no two

elements sum to 100.

The pairs in X that sum to 100 are

(2, 98), (6, 94), (10, 90), (14, 86), (18, 82), (22, 78), (26, 74), (30, 70), (34,

66), (38, 62), (42, 58), (46, 54).

Therefore, 

S = {2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50}

∴ The maximum possible number of elements in S be 13.

Let, x number of persons taking milk only.

According to the question

60 + 15 + (x - 5) + (50 - x) + x = 150

120 + x = 150

x = 150 - 120 = 30

∴ The required value is 30.

As y ∈ N, y can be 1, 2, 3, 4...
∴ x will be 1, 1/2, 1/3, 1/4...
⇒ 1 ∈ Q

• We know that, 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25
• Therefore the set A = {1, 4, 9, 16, 25...} can be written in set builder form as: 
  A = {x: x is the square of a natural number}

• According to this, x should be a number which is not equal to the number itself and there is no number which is not equal to the number itself.
• Therefore A is an empty set.

• A finite set is a set that has a finite number of elements.
• Since  x² – 25 = 0 has finite number of elements. So it is a finite set. 

From Venn-Euler's Diagram.
(A∪B)′∪(A′∩B)=A

Concept:

The null set is a subset of every set. (ϕ ⊆ A)

Every set is a subset of itself. (A ⊆ A)

The number of subsets of a set with n elements is 2ⁿ.

The number of proper subsets of a given set is 2ⁿ - 1

Calculation:

Statement I: If A = {x: x is an even natural number} and B = {y: y is a natural number}, A subset B.

A = {2, 4, 6, 8, 10, 12, ...} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...}.

It is clear that all the elements of set A are included in set B.

So, set A is the subset of set B.

Statement I is correct.

Statement II:  Number of subsets for the given set A = {5, 6, 7, 8} is 16.

Given: A = {5, 6, 7, 8}

The number of elements in the set is 4

We know that,

The formula to calculate the number of subsets of a given set is 2ⁿ

 = 2⁴ = 16

Number of subsets is 16

Statement II is incorrect.

Statement III: Number of proper subsets for the given set A = {5, 6, 7, 8} is 15.

The formula to calculate the number of proper subsets of a given set is 2ⁿ - 1

 = 2⁴ - 1

 = 16 - 1 = 15

The number of proper subsets is 15.

Statement III is correct.

∴ Statements I and III are correct.

See the following Venn diagram
n(I) = 29 + 23 = 52
n(F) = 100 − 52 = 48
n(M ∪ D) = n(M) + n(D) − n(M∩D)
24 = 23 + 4 − n(M∩D)
∴ n(M ∩ D) = 3
∴ n(W ∩ D) = 4 − 3 = 1

• Every set has the empty set as a subset. So if a set has 1 element, like {0}, then it will have 2 subsets: itself and the empty set, which is denoted by { }.
• So, if a set has only one subset, then this set must be the empty set.