What is the correct symbolic representation for "True if both are true, else false"?

The statement "True if both are true, else false" succinctly describes the logical operation known as the conjunction, which is represented by the symbol for "and" (Λ).

In terms of symbolic logic, the conjunction pΛq evaluates to true only when both propositions p and q are true. This means that if either p or q is false, or if both are false, the overall expression pΛq will also be false. Therefore, it directly captures the meaning of the statement provided.

In contrast, the other symbols represent different logical operations: the disjunction (V) indicates that at least one proposition must be true for the expression to be true, the implication (->) signifies a conditional relationship where the truth of one proposition guarantees the truth of another, and the negation (~) indicates the opposite truth value of a single proposition. These do not fulfill the requirement of being true only when both propositions are true. Thus, pΛq is indeed the correct representation for the provided statement.