Archer Sentences: Answers

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1. Every archer has arrows.

   ∀x[Archer(x) → HasArrows(x)]

2. Some archer has no arrows.

   ∃x[Archer(x) ∧ ¬HasArrows(x)]

3. Every archer hits a target.

   ∀x[Archer(x) → ∃y[Target(y) ∧ Hits(x,y)]]

4. Every target is hit by some archer.

   ∀y[Target(y) → ∃x[Archer(x) ∧ Hits(x,y)]]

5. Every archer hits a different target
   (in the sense that each archer hits only their own target and no other target)
   (get ready now…)

   ∀x[Archer(x) → ∃!y[Target(y) ∧ Hits(x,y)]]

   ∧ ¬∃x₁∃x₂∃y[Archer(x₁) ∧ Archer(x₂) ∧ x₁≠x₂ ∧ Target(y) ∧ Hits(x₁,y) ∧ Hits(x₂,y)]

6. No archer hits more than one target.
   (if you could read #5, then this one should be easy :-)

   ¬∃x∃y₁∃y₂[Archer(x) ∧ Target(y₁) ∧ Target(y₂) ∧ y₁≠y₂ ∧ Hits(x,y₁) ∧ Hits(x,y₂)]