Which of the following is a logically valid sentence? To show that something is not a valid sentence, it is sufficient to produce an interpretation in which the sentence fails to be true. The method described in the preceding problem is very helpful here-draw a directed graph representing a binary relation R in which the sentence is false. To show something is a valid sentence, try to give a cogent argument in words why it must be true in every interpretation. Incidentally, it is usual to require in the definition of the semantics that the domain be nonempty, which means, for example, that VxP(x) xP(x) is valid. (It would not be if we allowed empty domains.) (a) VxVyR(x,y) \xR(x,x). (b) VxyR(x,y) 3x R(x,x). (c) EyVxR(x,y) VxyR(x,y). (d) Vxy R(x, y) xVyR(x, y).