U= { x | x is the name of one of the states of in the us}
a = {x | x ∈ u and x begins with the letter a}
i = {x | x ∈ u and x begins with the letter i}
p = {x | x ∈ u and x begins with the letter p}
n = {x | x ∈ u and x begins with the letter n}
o = { x | x ∈ u and x begins with the letter o}

find:
n(p')
n(a' ∪ n)
n(i' ∩ o')
n(p∩n)

!

Step-by-step explanation:

We can start with some helpful facts:

and all subsets are mutually exclusive.

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n(P') = n(U) -n(P) = 50 -1 = 49

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n(A' ∪ N) = n(A') = n(U) -n(A) = 50 -4 = 46

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n(I' ∩ O') = n(U) -n(I) -n(O) = 50 -4 -3 = 43

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n(P∩N) = 0 . . . . no states begin with the letter P and also with the letter N.

For reference, here is a list of the relevant states:

{Alabama, Alaska, Arizona, Arkansas, Idaho, Illinois, Indiana, Iowa, Nebraska, Nevada, NewHampshire, NewJersey, NewMexico, NewYork, NorthCarolina, NorthDakota, Ohio, Oklahoma, Oregon, Pennsylvania}

this doesn't make sense lol