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 :

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

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

• 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.