X is said to entail sentence Y. If whenever X is true, Y is also TRUE. Which of the following is/are correct if X entails Y?

The correct options are:
1. If x then y
2. x → y
3. x ∧ ¬y is false

Step-by-step Explanation:

Logical entailment, denoted as $X\models Y$, means that in every possible world or truth assignment where $X$ is true, $Y$ must also be true. In other words, it is impossible for $X$ to be true while $Y$ is false.

Let us analyze each of the correct statements to understand why they follow from this definition:

1. "If x then y" and "x → y":
The implication statement "If x then y", represented symbolically as $x\to y$, asserts that if the antecedent $x$ holds, then the consequent $y$ must also hold. By definition, if $X$ entails $Y$, then whenever $X$ (or $x$) is true, $Y$ (or $y$) is also true. Thus, the implication $x\to y$ is logically valid under all circumstances where $X$ entails $Y$.

2. "x ∧ ¬y is false":
The statement $x\wedge \neg y$ represents the condition where $x$ is true AND $y$ is false (since $\neg y$ is the negation of $y$). As established by the definition of entailment, it is impossible for $x$ to be true while $y$ is false. Therefore, the combination $x\wedge \neg y$ can never happen (it is always false).

3. Why "If y then x" is incorrect:
Entailment is a one-way relationship. Just because $X$ being true guarantees that $Y$ is true, it does not mean that $Y$ being true guarantees $X$ is true. For example, "It is raining" ($X$) entails "The ground is wet" ($Y$). However, "The ground is wet" ($Y$) does not entail "It is raining" ($X$), because the ground could be wet for another reason (like a sprinkler).