Question

Please refer to the image here. ?

Answer 1

`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`

![enter image source here]
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And, `n(P' nn Q')=x`

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

`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`

`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'`

`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`