Answer to Question #211837 in Discrete Mathematics for Jaguar

Question

Question 22

Consider the following statement:

∀x  Z, [(2x + 4 > 0)  (4 - x

2 ≤ 0)]

The negation of the above statement is:

¬[∀x  Z, [(2x + 4 > 0)  (4 - x

2 ≤ 0)]]

≡ ∃x  Z, ¬[(2x + 4 > 0)  (4 - x

2 ≤ 0)]

≡ ∃x  Z, [¬(2x + 4 > 0) ∧ ¬(4 - x

2 ≤ 0)]

≡ ∃x  Z, [(2x + 4 ≤ 0) ∧ (4 - x

2 > 0)]

1. True

2. False

Question 23

Consider the statement

If n is even, then 4n2

- 3 is odd.

The contrapositive of the given statement is:

If 4n2

- 3 is odd, then n is even.

1. True

2. False

Answer 1

Question 22

Let us consider the statement $∀x \in\Z, [(2x + 4 > 0) \lor(4 - x^2 ≤ 0)]$

The negation of the above statement is:

$\neg[∀x \in\Z, [(2x + 4 > 0) \lor(4 - x^2 ≤ 0)]]≡ \exists x \in\Z, \neg[(2x + 4 > 0) \lor(4 - x^2 ≤ 0)]≡ \exists x \in\Z, [\neg(2x + 4 > 0) \land\neg(4 - x^2 ≤ 0)]≡ \exists x \in\Z, [(2x + 4 \le 0) \land(4 - x^2 >0)]$

Answer: 1. True

Question 23

Let us consider the statement "If $n$ is even, then $4n^2- 3$ is odd". The contrapositive law is $p\to q≡\neg q\to\neg p.$ Let $p=$ " $n$ is even", $q=$ " $4n^2- 3$ is odd". Then the contrapositive of the given statement is "If $4n^2- 3$ is even, then $n$ is odd".

Answer: 2. False