CS 281 Solutions to Homework 4

pp. 33 - 36

6 a) Some student (in this class) has taken some CS course.
d) Some CS course has been taken by every student.
e) Every CS course has been taken by some student.

10 a) $\exists x \exists y Q(x,y)$
b) $\forall x \forall y \neg Q(x,y)$ or $\neg \exists x \exists y Q(x,y)$
c) $\exists x [Q(x,J) \wedge Q(x,W)]$
d) $\forall y \exists x Q(x,y)$
e) $\exists x \exists z [Q(x,J) \wedge Q(z,J) \wedge x \neq z]$
Here $z$ represents a student.

12 a) $\forall x F(x, Fred)$
b) $\forall x F(Evelyn, x)$
c) $\forall x \exists y F(x,y)$
d) $\neg \exists x \forall y F(x,y)$
e) $\forall x \exists y F(y,x)$
h) $\exists x [\forall y F(y,x) \wedge \forall z [\forall w F(w,z) \rightarrow x = z]]$
i) $\neg \exists x F(x,x)$
j) $\exists x \exists y [F(x,y) \wedge x \neq y \wedge \forall z [(F(x,z) \wedge z \neq x) \rightarrow y = z]]$

14 d) $\forall x \neg C(x, Bob)$ or $\neg \exists x C(x, Bob)$
k) $\exists x [I(x) \wedge \forall y [ y \neq x \rightarrow \neg C(x, y)]]$
m) $\exists x \forall y C(x, y)$

28. Below predicates are used without any explanation. But what they represent should be obvious.

a) $\neg \forall x M(x)$
Some students don't like mathematics.
b) $\neg \exists x \neg C(x)$
Every student has seen a computer.
c) $\neg \exists x \forall y T(x,y)$
There is no student who has taken every mathematics course.
d) $\neg \exists x \forall z \exists y V(x,y,z)$
Here $V(x,y,z)$ means that a student $x$ has been in room $y$ of building $z$.
No student has been in any room of some building.

32 a) $\forall x [P(x) \rightarrow Q(x)]$
b) $\exists x [R(x) \wedge \neg Q(x)]$
c) $\exists x [R(x) \wedge \neg P(x)]$