6.3. Solutions Predicate Logic#

Solutions to Section 2.4

Solution to Exercise (1†)

  1. \(\exists x (P(x))\)

  2. \(\forall x(\lnot P(x)\vee \lnot Q(x))\)

  3. \(\exists z(P(z) \wedge \lnot Q(z))\)

  4. \((\exists x (\lnot P(x)))\vee (\exists y (\lnot Q(y)))\)

  5. \(\exists x \forall y \lnot P(x,y)\)

  6. \(\forall x (\lnot R(x)\vee \exists y \lnot S(x,y))\)

  7. \(\forall y((\lnot P(y)\wedge Q(y)) \vee (P(y) \wedge \lnot Q(y)))\)

  8. \(\exists x (P(x) \wedge (\forall y \lnot Q(x,y)))\)

Solution to Exercise (8†)

We use the predicates \( Ball (x)\) for \(x\) is a ball and \( Have (x,y)\) for \(x\) must have \(y\). We also use a constant \( you \) to represent you. \( \exists x (*Ball*(x) \wedge \forall y (x \neq y \rightarrow \lnot *Ball*(y)) \wedge *Have*(*you*, x)) \)

Solution to Exercise (13†)

Using the predicates \( Person (x)\) for \(x\) is a person, \( Question (x)\) for \(x\) is a question, \( Answer (x)\) for \(x\) is an answer, and \( Has (x,y)\) for \(x\) has \(y\). Two different interpretations are: \( \exists x (*Person*(x) \wedge \forall y (*Question*(y) \rightarrow (\exists z (*Answer*(z) \wedge *Has*(x,z))))) \) In other words, there is a single person who has the answers to all questions. \( \forall x (*Question*(x) \rightarrow \exists y \exists z (*Person*(y) \wedge *Answer*(z) \wedge *Has*(y,z))) \) In other words, every question has an answer and some person knows it (but different people might know the answer to different questions).