Question:
Which of the following statements is NOT true? (The names of the predicates are intuitive.)
Which of the following statements is NOT true? (The names of the predicates are intuitive.)
Show Hint
"Every A is a B" translates using implication (\(A(x) \Rightarrow B(x)\)), never a conjunction (\(A(x) \wedge B(x)\)).
Updated On: Jul 22, 2026
- \(\forall x\, \forall y\; Classmate(x,y) \Rightarrow Classmate(y,x)\)
- \(\forall x\; Likes(x, Icecream) \Rightarrow \neg \exists x\; \neg Likes(x, Icecream)\)
- "Each king is a person" is equivalent to \(\forall x\; IsKing(x) \wedge IsPerson(x)\)
- "All humans are mortal" is equivalent to \(\forall x\; IsHuman(x) \Rightarrow IsMortal(x)\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question.
We need to check four statements written in first-order logic and find the one that is logically incorrect, meaning either the formula is false in general or it wrongly claims to be equivalent to the given English sentence.
Step 2: Key Formula or Approach.
The standard rule for translating an English sentence of the form "every A is a B" or "each A is a B" into first-order logic is to use an implication: \(\forall x\; A(x) \Rightarrow B(x)\), meaning "for every x, if x is an A, then x is a B". Using a conjunction \(\wedge\) instead of an implication changes the meaning completely, since \(\forall x\; A(x) \wedge B(x)\) claims every single object in the whole domain is both an A and a B, which is a much stronger and usually false statement.
Step 3: Detailed Explanation.
Check option (A): the classmate relation is naturally symmetric, if x is a classmate of y then y is a classmate of x, so \(\forall x \forall y\; Classmate(x,y) \Rightarrow Classmate(y,x)\) correctly captures this and is a true statement. Check option (B): let \(A\) denote the sentence \(\forall x\; Likes(x, Icecream)\). The expression \(\neg \exists x\; \neg Likes(x, Icecream)\) is the textbook logical equivalent of \(\forall x\; Likes(x, Icecream)\), meaning it also equals \(A\). So the whole statement reads as \(A \Rightarrow A\), which is always true no matter what, a tautology, so option (B) is a true statement. Check option (C): the English sentence "Each king is a person" should translate as \(\forall x\; IsKing(x) \Rightarrow IsPerson(x)\), using an implication. But option (C) instead uses \(\forall x\; IsKing(x) \wedge IsPerson(x)\), a conjunction, which wrongly asserts that every object in the domain, kings and non-kings alike, is both a king and a person. This is not what the English sentence says, so this stated equivalence is false, making option (C) NOT true. Check option (D): "All humans are mortal" correctly translates as \(\forall x\; IsHuman(x) \Rightarrow IsMortal(x)\), the standard implication form, so this is a true statement.
Step 4: Final Answer.
The statement that is NOT true is option (C), because it wrongly uses a conjunction where an implication is needed. \[ \boxed{\text{Option (C)}} \]
We need to check four statements written in first-order logic and find the one that is logically incorrect, meaning either the formula is false in general or it wrongly claims to be equivalent to the given English sentence.
Step 2: Key Formula or Approach.
The standard rule for translating an English sentence of the form "every A is a B" or "each A is a B" into first-order logic is to use an implication: \(\forall x\; A(x) \Rightarrow B(x)\), meaning "for every x, if x is an A, then x is a B". Using a conjunction \(\wedge\) instead of an implication changes the meaning completely, since \(\forall x\; A(x) \wedge B(x)\) claims every single object in the whole domain is both an A and a B, which is a much stronger and usually false statement.
Step 3: Detailed Explanation.
Check option (A): the classmate relation is naturally symmetric, if x is a classmate of y then y is a classmate of x, so \(\forall x \forall y\; Classmate(x,y) \Rightarrow Classmate(y,x)\) correctly captures this and is a true statement. Check option (B): let \(A\) denote the sentence \(\forall x\; Likes(x, Icecream)\). The expression \(\neg \exists x\; \neg Likes(x, Icecream)\) is the textbook logical equivalent of \(\forall x\; Likes(x, Icecream)\), meaning it also equals \(A\). So the whole statement reads as \(A \Rightarrow A\), which is always true no matter what, a tautology, so option (B) is a true statement. Check option (C): the English sentence "Each king is a person" should translate as \(\forall x\; IsKing(x) \Rightarrow IsPerson(x)\), using an implication. But option (C) instead uses \(\forall x\; IsKing(x) \wedge IsPerson(x)\), a conjunction, which wrongly asserts that every object in the domain, kings and non-kings alike, is both a king and a person. This is not what the English sentence says, so this stated equivalence is false, making option (C) NOT true. Check option (D): "All humans are mortal" correctly translates as \(\forall x\; IsHuman(x) \Rightarrow IsMortal(x)\), the standard implication form, so this is a true statement.
Step 4: Final Answer.
The statement that is NOT true is option (C), because it wrongly uses a conjunction where an implication is needed. \[ \boxed{\text{Option (C)}} \]
Was this answer helpful?
0
0
Top GATE DA Data Science and Artificial Intelligence Questions
- Consider performing depth-first search (DFS) on an undirected and unweighted graph G starting at vertex s. For any vertex u in G, d[u] is the length of the shortest path from s to u. Let (u, v) be an edge in G such that d[u] < d[v]. If the edge (u, v) is explored first in the direction from u to v during the above DFS, then (u, v) becomes a _____ edge.
- Three fair coins are tossed independently. T is the event that two or more tosses result in heads. S is the event that two or more tosses result in tails. What is the probability of the event T ∩ S?
- Consider the matrix \(M=\begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}\).
Which ONE of the following statements is TRUE ? - For any twice differentiable function ƒ: R → R, if at some x* ∈ R, f'(x*) = 0 and f'(x*) > 0, then the function f necessarily has a ______ at x = x*.
Note : R denotes the set of real numbers. - Match the items in Column 1 with the items in Column 2 in the following table :
Column 1 Column 2 (p) First In First Out (i) Stacks (q) Lookup Operation (ii) Queues (r) Last In First Out (iii) Hash Tables
View More Questions
Top GATE DA Artificial Intelligence Questions
- Which of the following algo is NOT an example of uninformed search?
- Which of the following algorithms is NOT an example of uninformed search?
- 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 \(Y\)? - Consider the game tree for a two-player turn-taking minimax game as shown in the figure. The value of a terminal node represents the utility of the game state if the game ends there. The numbers written next to the edges denote the strategies.
There are two players MAX and MIN. At any particular state of the game, MAX prefers to move to a state of maximum value. On the other hand, MIN prefers to move to a state of minimum value.
Suppose MAX starts the game at the root and has three strategies: 1, 2 and 3. Next, MIN plays and also has three strategies: 1, 2 and 3. The game ends there. Both players always take optimal strategies throughout the game.
At the root, the best strategy for MAX is __________. (Answer in integer) - Assume that a Creative (\(C\)) person will Succeed (\(S\)) if the person is also Disciplined (\(D\)), but will not succeed otherwise. Now, consider the following statements:
(i) \(C \wedge S \Leftrightarrow D\)
(ii) \(C \Rightarrow (S \Leftrightarrow D)\)
(iii) \(C \Leftrightarrow ((D \Rightarrow S) \vee \neg S)\)
Which of the following options is correct?
View More Questions
Top GATE DA Questions
- Consider performing depth-first search (DFS) on an undirected and unweighted graph G starting at vertex s. For any vertex u in G, d[u] is the length of the shortest path from s to u. Let (u, v) be an edge in G such that d[u] < d[v]. If the edge (u, v) is explored first in the direction from u to v during the above DFS, then (u, v) becomes a _____ edge.
- Three fair coins are tossed independently. T is the event that two or more tosses result in heads. S is the event that two or more tosses result in tails. What is the probability of the event T ∩ S?
- Consider the matrix \(M=\begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}\).
Which ONE of the following statements is TRUE ? - For any twice differentiable function ƒ: R → R, if at some x* ∈ R, f'(x*) = 0 and f'(x*) > 0, then the function f necessarily has a ______ at x = x*.
Note : R denotes the set of real numbers. - Match the items in Column 1 with the items in Column 2 in the following table :
Column 1 Column 2 (p) First In First Out (i) Stacks (q) Lookup Operation (ii) Queues (r) Last In First Out (iii) Hash Tables
View More Questions