Please refer to the image here. ?

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1 Answer
Mar 10, 2017

# n(P' nn Q) = 22-x #
# n(P nn Q) \ \ = 13+x #
# n(P nn Q') = 15-x #

Explanation:

Using just set theory we have:

We are given #n(epsilon)=50#

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And, #n(P' nn Q')=x#

enter image source here

Part (i); #n(P' nn Q)#

enter image source here
# n(P' nn Q)+n(P)+x=n(epsilon) #
# :. n(P' nn Q)+28+x=50 #
# :. n(P' nn Q)=22-x #

Part (ii); #P nn Q#

enter image source here
# n(P uu Q)=n(P)+n(Q)-n(P nn Q) #
# :. 50-x=28+35-n(P nn Q) #
# :. n(P nn Q)=13+x #

Part (iii); #P nn Q'#

enter image source here
# n(P nn Q')+n(Q)+x=n(epsilon) #
# :. n(P nn Q')+35+x=50 #
# :. n(P nn Q')=15-x #

Part (iv): Range
The min value for each of the above is 0;

# n(P' nn Q')=x ge 0 \ \ \ \ \ \ \ \ => x ge 0#
# n(P' nn Q) \ = 22-x ge 0 => x le 22 #
# n(P nn Q) \ \ \ = 13+x ge 0 => 13 +x ge -13 #
# n(P nn Q') \ = 15-x ge 0 => x le 15 #

Combining we get #0 le x le 15#