HirePro Set Theory Quiz-1

Solution: If we see the pattern here, the numbers are squares of natural numbers, such as:

1^{2.5} = 1

2^{2.5} = 1.414

3^{2.5} = 15.58

4^{2.5} = 32

And so on.

A = {x : x is the power of a number 2.5 }

Or we can write;

A = {x : x = n^{2.5} , where n ∈ N}

Solution

(A\cup B) can have max 9 elements if all elements in A are distinct from that of B
(B\cup C) can have max 10 elements if all elements in C are distinct from that of B
So the intersection between 2 sets can’t have elements more than the number of elements of smallest set.
So here, maximum possible elements is 9

Solution
A = {A,C,E,S }
B = {S,U,C,E }
C= { S, C,E}
So words that can be formed using 3 alphabets = 3! = 6

Solution
A = { 1, 2, 3, 5, 8, 13}
B = { 5, 7, 11, 13, 17, 18}
Common elements = { 5, 13}
2 elements are common between A and B
So A \times B and B \times A will have 2^{2} elements common
i.e. 4 elements

Solution
f(p) = 3 p^{2} - 4p + 10
f(-2) = 3 (-2)^{2} - 4(-2) + 10
= 12 + 8+ 10
= 30

Solution
Odd functions can be represented as f(x) + f(-x) = 0
For option (d)
f(x) = -x^{5} and f(-x) = ( (-x)^{5} = x^{5}
Adding both we get zero

Solution
n(S) = 60
n(C) = 45% of 60 = 0.45 \times 60 = 27
n((C \cap W ) = 10% of 60 = 0.1 \times 60 = 6
We need to find n(W)
We have n(S) = n(C) + n(W) - n((C \cap W )
60 = 27 + n(W) - 6
Solving,
n(W) = 39

Solution
n(L) = 30%
n(C) = 20%
n(L \cup C ) = n(S) - n( people owning neither)
= 100 - 55 = 45%
So 2 is incorrect
n(L \cup C )= n(L) + n(C) - n(L \cap C )
45 = 30 + 20 - n(L \cap C )
n(L \cap C ) = 5%
So 1 is incorrect.
Given that n(L \cap C ) = 1500
5 % of n(S) = 1500
n(S) = \frac{1500}{0.05} = 30,000
So 3 is correct

Solution
C is set of all elements in A that are not in B

Solution
The function, f(x), is a one to one function when one unique element from its domain will return each element of its range. This means that for every value of x, there will be a unique value of y or f(x).