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: Build a truth table for the four possible cases.
$X$ and $Y$ can each be $TRUE$ or $FALSE$, giving four combinations. "$X$ entails $Y$" rules out exactly one of these four: the case $X = TRUE, Y = FALSE$. The remaining three combinations ($TT$, $FT$, $FF$) are all allowed under entailment.

Step 2: Evaluate each option in all three allowed combinations.
Case $X=T, Y=T$: $X \Rightarrow Y$ is $T$; $X \wedge \neg Y = T \wedge F = F$, so "is FALSE" holds; "if X then Y" holds since both are true; "if Y then X" also happens to hold here.
Case $X=F, Y=T$: $X \Rightarrow Y$ is $T$ (false antecedent makes implication true); $X \wedge \neg Y = F \wedge F = F$, so "is FALSE" holds; "if X then Y" holds vacuously; "if Y then X" FAILS here, because $Y$ is true but $X$ is false.
Case $X=F, Y=F$: $X \Rightarrow Y$ is $T$; $X \wedge \neg Y = F \wedge T = F$, so "is FALSE" holds; "if X then Y" holds vacuously; "if Y then X" holds vacuously here (Y is false).

Step 3: Read off which statements hold in every allowed case.
$X \Rightarrow Y$: true in all three allowed cases, always holds under entailment.
$X \wedge \neg Y$ is FALSE: true in all three allowed cases, always holds under entailment.
"if X then Y": true in all three allowed cases (this phrase means the same thing as $X \Rightarrow Y$), always holds.
"if Y then X": this FAILS in the case $X=F, Y=T$, which is a case entailment does allow, so this statement is not guaranteed.

Step 4: Conclusion.
Only the case $X=F, Y=T$ is needed as a single counterexample to rule out option (D), since entailment permits that case but "if Y then X" breaks there. Options (A), (B), (C) hold in all cases entailment allows. \[ \boxed{\text{Options (A), (B) and (C) are correct, (D) is not}} \]