Conditional Statement Logic

The core of this question lies in understanding a conditional statement and its logical implications. We are given a main statement and four individual statements, which form pairs. We need to find a pair where the first statement logically leads to the second, and both statements are consistent with the original premise.

The main statement is: "If my mother cooks dinner, she garnishes extra cheese for me."

Let's break this down into logical propositions:

Let P represent: "my mother cooks dinner".

Let Q represent: "she garnishes extra cheese for me".

The main statement is a conditional, expressed in logic as: P $\rightarrow$ Q.

Analyzing Individual Statements

We are given four statements (A, B, C, D) that relate to P and Q:

• A. my mother cooked the dinner: This is equivalent to P.
• B. my mother did not cook dinner: This is the negation of P, denoted as ¬P.
• C. My food is garnished with extra cheese: This is equivalent to Q.
• D. My food is not garnished with extra cheese: This is the negation of Q, denoted as ¬Q.

Evaluating Pairs for Implication

The task is to find an ordered pair (X, Y) such that X $\rightarrow$ Y holds true, and both X and Y are logically consistent with P $\rightarrow$ Q. The correct answer option is given as DB.

Let's examine the pair (D, B):

• Statement D represents ¬Q ("My food is not garnished with extra cheese.").
• Statement B represents ¬P ("my mother did not cook dinner.").

The pair (D, B) suggests the implication: ¬Q $\rightarrow$ ¬P.

Contrapositive: Key to Consistency

In propositional logic, the contrapositive of a statement "If P, then Q" (P $\rightarrow$ Q) is "If not Q, then not P" (¬Q $\rightarrow$ ¬P).

A crucial rule in logic is that a conditional statement is logically equivalent to its contrapositive. This means that if P $\rightarrow$ Q is true, then ¬Q $\rightarrow$ ¬P must also be true, and vice versa. They are consistent with each other.

Since the main statement is P $\rightarrow$ Q, its contrapositive ¬Q $\rightarrow$ ¬P is logically implied and consistent.

Therefore, the pair DB (representing ¬Q $\rightarrow$ ¬P) accurately satisfies the conditions of the question.

Other potential pairs, like (B, D) representing ¬P $\rightarrow$ ¬Q, or (D, A) representing ¬Q $\rightarrow$ P, do not necessarily follow from the original statement P $\rightarrow$ Q because they are not logically equivalent forms like the contrapositive.