Question

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?

• A. If x then y
• B. x $\rightarrow$ y
• C. x $\land$ $\neg$y is false
• D. If y then x

Hint:

Remember the three equivalent ways to define entailment ($X \models Y$): 1. Whenever X is true, Y is also true. 2. The implication $X \rightarrow Y$ is a tautology (always true). 3. The statement $X \land \neg Y$ is a contradiction (always false).

Answer 1

The Correct Option is A, B, C

Step 1: Understanding the Question:
The question defines logical entailment ($X \models Y$) as "Whenever X is true, Y is also true." We need to identify which of the given options are equivalent to this definition.
Step 2: Detailed Explanation of Entailment:
The statement "$X$ entails $Y$" means that there is no possible world or interpretation where X is true and Y is false. Let's analyze the options based on this definition.
- (A) If x then y: This is the standard English phrasing for a logical implication. It directly matches the definition given: if X is true, then Y must be true. So, (A) is correct.
- (B) x $\rightarrow$ y: This is the formal logical notation for the material implication "if x, then y". The implication $X \rightarrow Y$ is false only when X is true and Y is false. The statement that $X \rightarrow Y$ is a valid statement (always true) is the formal definition of entailment. So, (B) is correct.
- (C) x $\land$ $\neg$y is false: This statement says that the situation "X is true AND Y is false" cannot happen (it is false or unsatisfiable). This is precisely the definition of entailment. If we can never have X true and Y false, then it must be that whenever X is true, Y is also true. So, (C) is correct.
- (D) If y then x: This represents the converse implication, $Y \rightarrow X$. Entailment is not symmetric. For example, "It is raining" (X) entails "The ground is wet" (Y). But "The ground is wet" (Y) does not entail "It is raining" (X) - a sprinkler could be on. So, (D) is not necessarily correct.
Step 3: Final Answer:
The correct statements that are equivalent to "X entails Y" are (A), (B), and (C).