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Question 1: Nested Quantifiers [True/False]........ 2 points (a) (1/2 point) \(\forall n \in \mathbb{N} \exists m \in \mathbb{N}: nm \ge 100\) (b) (1½ point) \(\forall n \in \mathbb{Z}^+ \exists m \in \mathbb{N}: nm \ge 100\) (c) (1½ point) \(\exists n \in \mathbb{N} \forall m \in \mathbb{N}: nm = n\) (d) (1½ point) \(\exists n \in \mathbb{Z}^+ \forall m \in \mathbb{N}: nm = n\) Question 2: Propositional Equivalences........ 4 points Show the following logically equivalence by developing a series of logical equivalences (no truth table): \(\neg (p \lor (\neg p \land q)) \equiv \neg p \land \neg q\)

          Question 1: Nested Quantifiers [True/False]........ 2 points
(a) (1/2 point) \(\forall n \in \mathbb{N} \exists m \in \mathbb{N}: n*m \ge 100\)
(b) (1½ point) \(\forall n \in \mathbb{Z}^+ \exists m \in \mathbb{N}: n*m \ge 100\)
(c) (1½ point) \(\exists n \in \mathbb{N} \forall m \in \mathbb{N}: n*m = n\)
(d) (1½ point) \(\exists n \in \mathbb{Z}^+ \forall m \in \mathbb{N}: n*m = n\)
Question 2: Propositional Equivalences........ 4 points
Show the following logically equivalence by developing a series of logical
equivalences (no truth table):
\(\neg (p \lor (\neg p \land q)) \equiv \neg p \land \neg q\)

Question 1: Nested Quantifiers [True/False]........ 2 points
(a) (1/2 point) ∀ n ∈ℕ∃ m ∈ℕ: n*m ≥ 100
(b) (1½ point) ∀ n ∈ℤ^+ ∃ m ∈ℕ: n*m ≥ 100
(c) (1½ point) ∃ n ∈ℕ∀ m ∈ℕ: n*m = n
(d) (1½ point) ∃ n ∈ℤ^+ ∀ m ∈ℕ: n*m = n
Question 2: Propositional Equivalences........ 4 points
Show the following logically equivalence by developing a series of logical
equivalences (no truth table):
(p ( p q)) ≡ p q

Added by Teresa T.

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