Understanding the Logical Statements

The core of the question is based on a main logical statement and four individual statements (A, B, C, D). We need to find an ordered pair of statements where the first statement logically implies the second, and both statements are consistent with the main statement.

Main Statement Analysis

The main statement is: "She is either a housewife or a working woman."

Let's represent this using propositional logic:

• Let $H$ represent "She is a housewife".
• Let $W$ represent "She is a working woman".

The main statement can be written as a disjunction (OR operation): $$H \lor W$$ This means that at least one of the conditions (being a housewife or being a working woman) must be true. It's possible both are true, or only one is true.

Individual Statement Representation

The individual statements are:

• A: She is a housewife (Symbolically: $H$)
• B: She is not a housewife (Symbolically: $\neg H$)
• C: She is a working woman (Symbolically: $W$)
• D: She is not a working woman (Symbolically: $\neg W$)

Identifying the Correct Pair (D, A)

The question asks for a pair (X, Y) such that X implies Y ($X \implies Y$) and both X and Y are consistent with the main statement ($H \lor W$). The provided correct option is DA, meaning statement D implies statement A.

Checking Implication: Does D imply A?

Statement D is $\neg W$ ("She is not a working woman").

Statement A is $H$ ("She is a housewife").

We need to check if $\neg W \implies H$ is true, given the main statement $H \lor W$.

Consider the main statement $H \lor W$. If we know that D ($\neg W$) is true, it means $W$ is false. Substituting $W=False$ into the main statement gives:

For this expression to be true (as the main statement must hold), $H$ must be true.

Therefore, if "She is not a working woman" ($\neg W$) is true, then "She is a housewife" ($H$) must also be true. The implication $\neg W \implies H$ holds true.

Checking Consistency

Now we check if both statements D ($\neg W$) and A ($H$) are consistent with the main statement ($H \lor W$).

If D ($\neg W$) is true and A ($H$) is true, we evaluate the main statement:

Substituting the truth values (H=True, W=False):

This evaluates to $True$. Since the result is true, the pair of statements (D and A) is consistent with the main statement.

Evaluating Other Options (Briefly)

Let's briefly consider why other pairs might not fit:

• AB: Does A imply B ($H \implies \neg H$)? No, being a housewife doesn't imply not being a housewife.
• CD: Does C imply D ($W \implies \neg W$)? No, being a working woman doesn't imply not being a working woman.
• AC: Does A imply C ($H \implies W$)? No, being a housewife does not necessarily mean she is also a working woman.

The pair DA fulfills both conditions: the implication holds, and the statements are consistent with the primary condition.