Test: Sets - Civil Engineering (CE) MCQ

The set S consists of the square of an integer less than 10.

Concept:
Between any two real numbers, there are infinitely many rationals and irrational.
{The numbers which are not rational are irrational}
Real = Rational ∪ Irrational

Explanation:
1. The set of all irrational numbers between √2 and √5 is an infinite set.
There are infinitely many rationals and irrational between any two real numbers.
 √2 and √5 are two irrational numbers and there are infinitely many rationals and irrationals between them.
⇒ The set of all irrational numbers between √2 and √5 is an infinite set.
So, statement 1 is true.

2. The set of all odd integers less than 100 is a finite set.
The set of all odd integers less than 100 is given by
{......,-5 ,-3 ,-1 ,1 ,3 ,5 ,.....,95, 97, 99}
this set is bounded above but not bounded below
So, It is an infinite set, and statement 2 is false.
∴ The correct option is (1).

Given:
Set A = {1, 2, 3, 4, 5}
Concept:
Subset - Set A is said to be a subset of Set B if all the elements of Set A are also present in Set B.

Solution:
A subset of A which contains element 2 but not 4 as follows,
{1, 2, 3, 5}, {1, 2, 5}, {1, 2, 3}, {1, 2}, {2}, {2, 3}, {2, 5}, {2, 3, 5}
The total numbers of the subset are 8.

Important point:

If a set has n elements then the power set of that set has 2n elements.

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 2^m - 1.
So, The number of proper subsets of A ∩  B = 2⁴ - 1 = 15
Hence, the correct option is 1

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.

Odd numbers less than 10 is {1, 3, 5, 7, 9}.

Let A = {1, 2} and B = {a, b}. The Cartesian product A x B = {(1, a), (1, b), (2, a), (2, b)} and the Cartesian product B x A = {(a, 1), (a, 2), (b, 1), (b, 2)}. This is not equal to A x B.

Two set are equal if and only if they have the same elements.

A subset R of the Cartesian product A x B is a relation from the set A to the set B.