Sets

More Topic from Mathematics

Question 1

A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P (A ∪ B) is 

  • 3/5

  • 0

  • 1

  • 2/5

Solution

C.

1

A = {4, 5, 6} , B = {1, 2, 3, 4} .
Obviously P (A ∪ B) = 1.

Question 2

If A, B and C are three sets such that A ∩ B = A∩ C and A ∪ B = A ∪ C, then

  • A = B

  • A = C

  • B = C

  • A ∩ B = φ

Solution

C.

B = C

A ∪ B = A ∪ C
⇒ n (A ∪ B) = n(A ∪ C)
⇒ n(A) + n(B) – n(A ∩ B)
= n(A) + n(C) – n(A ∩C)
n(B) = n(C)

Question 3

If X = {4n - 3n-1 : n ε N} and Y = {9(n-1):n εN}; where N is the set of natural numbers,then X U Y is equal to

  • N

  • Y-X

  • X

  • Y

Solution

D.

Y

We have X = {4n - 3n-1 : n ε N} 
X = {0,9,54,243,.....} [put n = 1,2,3....]
Y = {9(n-1):n ε N}
Y = {0,9,18,27,......}[Put n = 1,2,3....]
It is clear that 
X ⊂ Y
Therefore, X U Y = Y

Question 4

Let A and B be two events such that straight P space left parenthesis top enclose straight A union straight B end enclose right parenthesis space equals space 1 over 6 comma space straight P space left parenthesis straight A intersection straight B right parenthesis space equals space 1 fourth space and space straight P space left parenthesis top enclose straight A right parenthesis space equals space 1 fourth comma where top enclose straight A stands for complement of event A. Then events A and B are

  • equally likely and mutually exclusive

  • equally likely but not independent

  • independent but not equally likely

  • mutually exclusive and independent

Solution

C.

independent but not equally likely

straight P space open parentheses top enclose straight A space union straight B end enclose close parentheses space equals space 1 over 6 comma space straight P space left parenthesis straight A space intersection straight B right parenthesis space equals space 1 fourth space and space straight P space left parenthesis top enclose top enclose straight A end enclose right parenthesis space equals space 1 fourth
rightwards double arrow space straight P space left parenthesis straight A union straight B right parenthesis space equals space 5 divided by 6 space straight P space left parenthesis straight A right parenthesis space equals space 3 divided by 4
Also P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ P(B) = 5/6 – 3/4 + 1/4 = 1/3
P(A) P(B) = 3/4 – 1/3 = 1/4 = P(A ∩ B)
Hence A and B are independent but not equally likely.
Question 5

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A × B having 3 or more elements is

  • 256

  • 220

  • 219

  • 211

Solution

C.

219

Given, n(A) =2, n(B) = B
The number of subsets of AXB having 3 or more elements,
=straight C presuperscript 8 subscript 3 space plus to the power of 8 straight C subscript 4 space plus space..... plus to the power of 8 straight C subscript 8
space equals space 2 to the power of 8 space minus to the power of 8 straight C subscript 0 space minus to the power of 8 straight C subscript 1 space minus space to the power of 8 straight C subscript 2
space equals space 256 minus 1 minus 8 minus 28 space equals space 219