Discrete Mathematics
1.5.15
(a)∀xP(x), x:a computer science student in this class, P(x):x needs a
course in discrete mathematics.
(c)∀x∃yP(x,y), x:a student in this class, y:a computer science course,
P(x,y):x has taken y.
1.5.28
(a)This statement is true because we can let y be a function of x, and
y=x^2. So for every x there exists a number y satisfying the statement.
(b)This statement is false because when x<0, there isn’t any number
whose square is negative.
(h)We can solve these equation and find that there isn’t any solution
(x,y), so this statement is false.
1.5.38
(a)¬(∀xP(x))≡∃x¬P(x), x:a student in this class, P(x):x likes mathematics.
Negation: There exists a student in this class who doesn’t like
mathematics.
(b)¬(∃xP(x))≡∀x¬P(x), x: a student in this class, P(x):x has never seen a
computer.
Negation: Every student in this class has seen a computer.
(c)¬(∃x∀yP(x,y))≡∀x¬(∀yP(x,y))≡∀x∃y¬P(x,y), x: a student in this class,
y:a mathematics course offered at this school, P(x,y):x has taken y.
Negation: For every student in this class there exists a mathematics
course offered at this school which he or she hasn’t taken.
(d)¬(∃x∀y∃zP(x,y,z))≡∀x¬(∀y∃zP(x,y,z))≡∀x∃y¬(∃zP(x,y,z))≡∀x∃y∀z¬P
(x,y,z), x: a student in this class, y: a building on campus, z: a
room,P(x,y,z):x has been in y’z
Negation: For every student in this class there exists a building on
campus, and he or she hasn’t been in any room of it.
