MACM 101 — Discrete Mathematics I
Exercises on Predicates and Quantifiers.
Due:
Friday, October 14th (at the beginning
of the class)
Reminder: the work you submit must be your own. Any collabora
tion and consulting outside resources must be explicitly mentioned
on your submission.
Please, use a pen.
30 points will be taken off for pencil written
work.
1. Determine the truth value of each of these statements if the universe
of each variable consists of all real numbers.
(a)
∃
x
∃
y
((
x
+ 2
y
= 2)
∧
(2
x
+ 4
y
= 5));
(b)
∀
x
∀
y
∃
z
(
x
2
+
y
2
=
z
2
).
2. Use predicates and quantifiers to express this statement
“Every student has chatted with at least one other student in at least
one chat group.”
3. Find a counterexample, if possible, to this quantified statement, where
the universe for all variables consists of all integers
∀
x
∃
y
(
y
2
=
x
)
.
4. Rewrite the following statement so that negations appear only within
predicates (that is, no negation is outside a quantifier or an expression
involving logical connectives)
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 Spring '08
 PEARCE
 Logic, Real Numbers, logical connectives, Truth value, chat group

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