Let N=250 be the total members of the society voted for X,Y,Z. Let n(X),n(Y),n(Z) denote the number of candidates voted for X,Y,Z respectively. Then,
N250=n(X∪Y∪Z)+n(X∪Y∪Z)′(Voted + Not voted)=n(X)+n(Y)+n(Z)−n(X∩Y)−n(X∩Z)−n(Y∩Z)+n(X∩Y∩Z)+n(X∪Y∪Z)′=n(X)+n(Y)+n(Z)−n(X∩Y)−n(X∩Z)−n(Y∩Z)+n(X∪Y∪Z)′(1)(Since each member may vote for either one or two candidates n(X∩Y∩Z)=0)
12 voted for X and Y, 14 voted for X and Z, i.e.,n(X∩Y)n(X∩Z)=12(2)=14(3)59 voted for Y only and 37 voted for Z only, i.e.,n(Y)−n(X∩Y)−n(Y∩Z)n(Z)−n(X∩Z)−n(Y∩Z)=59(4)=37(5)147 voted for X or Y or both but not for Z 102 voted for Y or Z or both but not for X, i.e.,n(X)+n(Y)−n(X∩Y)−n(X∩Z)−n(Y∩Z)n(Y)+n(Z)−n(Y∩Z)−n(X∩Z)−n(X∩Y)=147(6)=102(7)
Adding (4) and (5), we get
n(Y)+n(Z)−n(X∩Y)−n(X∩Z)−2⋅n(Y∩Z)=96(8)
Subtracting (8) from (7), we get n(Y∩Z)=6.
From (4), (5), (6)
n(Y)n(Z)n(X)=59+n(X∩Y)+n(Y∩Z)=59+12+6=77=37+n(X∩Z)+n(Y∩Z)=37+14+6=57=147−n(Y)+n(X∩Y)+n(X∩Z)+n(Y∩Z)=147−77+12+14+6=102
i)
ii) Number of candidates who did not vote = n(X∪Y∪Z)′
From (1),
n(X∪Y∪Z)′=250−(n(X)+n(Y)+n(Z)−n(X∩Y)−n(X∩Z)−n(Y∩Z))=250−(102+77+57−12−14−6)=46
iii) Number of members voted for X only = n(X)−n(X∩Y)−n(X∩Z)=102−12−14=76
iv) Number of votes for X only = 76,
Number of votes for Y only = 59,
Number of votes for Z only = 37.
Therefore, X won the election.