Let the predicates D(x, y) mean “team x defeated team y” and P(x, y) mean “team x has played team y”. The quantified formula for the statement that there is a team that has beaten every team it has played, is :

Question

Question

This question was previously asked in

NIELIT Scientific Assistant IT 5 Dec 2021 Official Paper

Answer 1

The correct answer is: option 1: ∃x∀y (P(x, y) → D(x, y))

Concept:

We are given two predicates:

Target Statement: "There is a team that has beaten every team it has played."

This can be broken down logically as:

There exists a team x such that for all teams y, if x has played y, then x has defeated y.

This is formally written as:

∃x ∀y (P(x, y) → D(x, y))

Explanation of options:

Option 1 – ∃x∀y (P(x, y) → D(x, y)): ✅ Correct. Matches the target statement perfectly.
Option 2 – ∀x∃y (P(x, y) → D(x, y)): ❌ Says every team has at least one team it has beaten if it has played it — weaker statement.
Option 3 – ∀y∃x (P(x, y) → D(x, y)): ❌ Misplaces quantifiers — for every opponent, some team beat them — not what is asked.
Option 4 – ∃x∀y (D(x, y) → P(x, y)): ❌ Illogical — says if x defeated y, then x played y — always true but not relevant.

Hence, the correct answer is: option 1: ∃x∀y (P(x, y) → D(x, y))