Question:
Which of the following NOT true? (The name of the predicate are intuitive)
Which of the following NOT true? (The name of the predicate are intuitive)
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A common mistake in first-order logic is translating "All A are B" into $\forall$x (A(x) $\wedge$ B(x)). This is wrong. It implies everything is A and B. The correct form uses implication: $\forall$x (A(x) $\Rightarrow$ B(x)). Conversely, "Some A are B" is translated using conjunction: $\exists$x (A(x) $\wedge$ B(x)).
Updated On: Feb 23, 2026
- $\forall$x likes (x, Ice-cream) $\Rightarrow$ $\neg$$\exists$x$\neg$ likes (x, Ice-cream)
- $\forall$x $\forall$y classmate (x, y) $\Rightarrow$ Classmate (y, x)
- "All humans are mortal" is equivalent to $\forall$x is human (x) $\Rightarrow$ Ismortal(x)
- "Each King is a person" is equivalent to $\forall$ Isking (x) $\wedge$ Isperson(x)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We need to evaluate four statements involving first-order logic and identify the one that is NOT true or logically incorrect.
Step 2: Detailed Explanation:
Let's analyze each option:
(A) $\forall$x likes (x, Ice-cream) $\Rightarrow$ $\neg$$\exists$x$\neg$ likes (x, Ice-cream)
This statement relates the universal quantifier ($\forall$) and the existential quantifier ($\exists$).
The left side, $\forall$x P(x), means "For all x, P(x) is true." (Everyone likes Ice-cream).
The right side, $\neg$$\exists$x$\neg$P(x), means "It is not the case that there exists an x for which P(x) is not true." (There is no one who does not like Ice-cream).
These two statements are logically equivalent. This is a fundamental identity in predicate logic. Thus, statement (A) is true.
(B) $\forall$x $\forall$y classmate (x, y) $\Rightarrow$ Classmate (y, x)
This statement is asserting that the `classmate` relation is symmetric. If x is a classmate of y, then y is a classmate of x. Based on the intuitive meaning of "classmate", this property holds. Thus, statement (B) is true.
(C) "All humans are mortal" is equivalent to $\forall$x is human (x) $\Rightarrow$ Ismortal(x)
This is the standard and correct way to translate a universally quantified statement of the form "All A's are B's" into first-order logic. It reads: "For any x, if x is a human, then x is mortal." This correctly captures the meaning of the English sentence. Thus, statement (C) is true.
(D) "Each King is a person" is equivalent to $\forall$x Isking (x) $\wedge$ Isperson(x)
This translation is incorrect. The logical statement $\forall$x (Isking(x) $\wedge$ Isperson(x)) reads as "For all x, x is a king AND x is a person." This means that everything in the universe is both a king and a person, which is clearly false and does not match the meaning of the original sentence.
The correct translation for "Each King is a person" (or "All kings are persons") is, similar to option (C), $\forall$x (Isking(x) $\Rightarrow$ Isperson(x)), which means "For any x, if x is a king, then x is a person."
Since the provided translation is incorrect, statement (D) is NOT true.
Step 3: Final Answer:
Statement (D) provides an incorrect logical representation of the given English sentence. Therefore, it is the statement that is not true.
We need to evaluate four statements involving first-order logic and identify the one that is NOT true or logically incorrect.
Step 2: Detailed Explanation:
Let's analyze each option:
(A) $\forall$x likes (x, Ice-cream) $\Rightarrow$ $\neg$$\exists$x$\neg$ likes (x, Ice-cream)
This statement relates the universal quantifier ($\forall$) and the existential quantifier ($\exists$).
The left side, $\forall$x P(x), means "For all x, P(x) is true." (Everyone likes Ice-cream).
The right side, $\neg$$\exists$x$\neg$P(x), means "It is not the case that there exists an x for which P(x) is not true." (There is no one who does not like Ice-cream).
These two statements are logically equivalent. This is a fundamental identity in predicate logic. Thus, statement (A) is true.
(B) $\forall$x $\forall$y classmate (x, y) $\Rightarrow$ Classmate (y, x)
This statement is asserting that the `classmate` relation is symmetric. If x is a classmate of y, then y is a classmate of x. Based on the intuitive meaning of "classmate", this property holds. Thus, statement (B) is true.
(C) "All humans are mortal" is equivalent to $\forall$x is human (x) $\Rightarrow$ Ismortal(x)
This is the standard and correct way to translate a universally quantified statement of the form "All A's are B's" into first-order logic. It reads: "For any x, if x is a human, then x is mortal." This correctly captures the meaning of the English sentence. Thus, statement (C) is true.
(D) "Each King is a person" is equivalent to $\forall$x Isking (x) $\wedge$ Isperson(x)
This translation is incorrect. The logical statement $\forall$x (Isking(x) $\wedge$ Isperson(x)) reads as "For all x, x is a king AND x is a person." This means that everything in the universe is both a king and a person, which is clearly false and does not match the meaning of the original sentence.
The correct translation for "Each King is a person" (or "All kings are persons") is, similar to option (C), $\forall$x (Isking(x) $\Rightarrow$ Isperson(x)), which means "For any x, if x is a king, then x is a person."
Since the provided translation is incorrect, statement (D) is NOT true.
Step 3: Final Answer:
Statement (D) provides an incorrect logical representation of the given English sentence. Therefore, it is the statement that is not true.
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