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McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Exercise Sheet 2
COMP SCI 1FC3 — Mathematics for Computing
16 January 2009
Exercise 2.1 — Predicates, formal and English
From the textbook section 1.3, do
at least
exercises 10–14, 25, 28, 30. 38–42 (pp. 47–49).
Exercise 2.2 — Predicate Logic Equivalences
Using logical equivalences from the text book, prove the following:
(a)
(∃
x
(
P
(
x
))) ∨ (∃
x
(
Q
(
x
))) ≡ ∃
x
(
P
(
x
) ∨
Q
(
x
))
Using semantic arguments (and possibly logical equivalences), prove the following (assuming
x
does not occur free
in
A
, and that the domain of quanti±cation is nonempty):
(b)
(∀
x
(
P
(
x
))) ∧
A
≡ ∀
x
(
P
(
x
) ∧
A
)
(c)
(∃
x
(
P
(
x
))) ∧
A
≡ ∃
x
(
P
(
x
) ∧
A
)
(d)
∀
x
(
A
→
P
(
x
)) ≡
A
→ ∀
x
(
P
(
x
))
(e)
∃
x
(
A
→
P
(
x
)) ≡
A
→ ∃
x
(
P
(
x
))
(f)
∀
x
(
P
(
x
) →
A
) ≡ (∀
x
(
P
(
x
))) →
A
)
(g)
∃
x
(
P
(
x
) →
A
) ≡ (∃
x
(
P
(
x
))) →
A
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