38 Let sets X Y and Z be characteristic sets of formulas Px

(3.8) Let sets X, Y and Z be characteristic sets of formulas P(x), Q(x) and R(x) respectively. For each possible region of the Venn diagram of X, Y and Z give a compound formula (with atomic formulas P, Q and R) that has that region as its characteristic set.

Solution

Draw three overlapping circles to represent the three sets

Write down the elements in the intersection X Y Z=PQR

Write down the remaining elements in the intersections: X Y=PQ, Y Z=QR and X Z=PR

Write down the remaining elements in the respective sets.