Question

Sentence \(X\) is said to entail Sentence \(Y\) if whenever \(X\) is \(TRUE\), \(Y\) also must hold \(TRUE\).
Which of the following statements is/are correct if \(X\) entails \(Y\)?

• A. \(X \Rightarrow Y\)
• B. \(X \wedge \neg Y\) is \(FALSE\)
• C. if \(X\) then \(Y\)
• D. if \(Y\) then \(X\)

Hint:

Entailment rules out only \(X = TRUE, Y = FALSE\). Check each statement against this and note that "if Y then X" is the converse, which is not guaranteed.

Answer 1

The Correct Option is A, B, C

Step 1: Restate the definition of entailment in logic form.
\(X\) entails \(Y\) means: whenever \(X\) is \(TRUE\), \(Y\) must also be \(TRUE\). There is no case where \(X\) is \(TRUE\) and \(Y\) is \(FALSE\) at the same time.

Step 2: Check option (A), \(X \Rightarrow Y\).
The logical implication \(X \Rightarrow Y\) is defined to be \(FALSE\) only in the one case where \(X\) is \(TRUE\) and \(Y\) is \(FALSE\), and \(TRUE\) in every other case. This is exactly the same condition as "whenever \(X\) is \(TRUE\), \(Y\) is \(TRUE\)". So \(X \Rightarrow Y\) correctly captures entailment. Option (A) is correct.

Step 3: Check option (B), \(X \wedge \neg Y\) is \(FALSE\).
\(X \wedge \neg Y\) is \(TRUE\) exactly when \(X\) is \(TRUE\) and \(Y\) is \(FALSE\) simultaneously, this is the one forbidden case under entailment. Saying \(X \wedge \neg Y\) is \(FALSE\) is just another way of saying this forbidden case never happens, which is the definition of entailment itself. Option (B) is correct, and in fact it is logically identical to \(X \Rightarrow Y\).

Step 4: Check option (C), "if \(X\) then \(Y\)".
In plain English, "if \(X\) then \(Y\)" is the natural-language phrasing of the material implication \(X \Rightarrow Y\), it means the same thing: whenever \(X\) holds, \(Y\) must hold too. Option (C) is correct, it is just a restatement of (A) in words.

Step 5: Check option (D), "if \(Y\) then \(X\)".
This is the converse of the implication, \(Y \Rightarrow X\). Entailment only guarantees that \(X\) being \(TRUE\) forces \(Y\) to be \(TRUE\), it says nothing about what happens if \(Y\) is \(TRUE\), \(X\) could very well be \(FALSE\) in that case. For example, "it is raining" entails "the ground is wet", but "the ground is wet" does not entail "it is raining" (the ground could be wet for other reasons). So option (D) does not follow from \(X\) entailing \(Y\), it is incorrect.

Final Answer:
Options (A), (B) and (C) are all correct restatements of "\(X\) entails \(Y\)", while (D) is the converse and is not implied. \[ \boxed{\text{Options (A), (B), (C)}} \]