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 :

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NIELIT Scientific Assistant CS 5 Dec 2021 Official Paper
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  1. ∃x∀y (P(x, y) → D(x, y))
  2. ∀x∃y (P(x, y) → D(x, y))
  3. ∀y∃x (P(x, y) → D(x, y))
  4. ∃x∀y (D(x, y) → P(x, y))

Answer (Detailed Solution Below)

Option 1 : ∃x∀y (P(x, y) → D(x, y))
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Explanation of Correct Answer - khautorepair.com

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

key-point-image Key Points

  • The statement ∃x∀y (P(x, y) → D(x, y)) translates to "There exists a team x such that for every team y, if team x has played team y, then team x has defeated team y". This matches the given statement: "There is a team that has beaten every team it has played".
  • The key part of this statement is the implication (P(x, y) → D(x, y)), which ensures that the team has defeated every other team it has played against.
  • The existential quantifier ∃x indicates that at least one such team exists.

additional-information-image Additional Information

  • Option 2 (∀x∃y (P(x, y) → D(x, y))) means "For every team x, there exists a team y such that if team x has played team y, then team x has defeated team y", which is not the same as the given statement.
  • Option 3 (∀y∃x (P(x, y) → D(x, y))) means "For every team y, there exists a team x such that if team x has played team y, then team x has defeated team y", which does not match the given statement either.
  • Option 4 (∃x∀y (D(x, y) → P(x, y))) means "There exists a team x such that for every team y, if team x has defeated team y, then team x has played team y", which is not relevant to the given statement.
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