MMW (Assessment 2) PDF

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Assessment 21. Write each statement in words. Let p: The plane is on time. Let q: The sky is clear.(a) p ˄ (¬ q)The plane is on time and the sky is not clear.(b) q → (p ˅ ¬ p)If the sky is clear, either the plane is on time or not.(c) p ↔ qThe plane is on time if and only if the sky is clear.→ ˅ ¬ ↔...

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Assessment 2 1. Write each statement in words. Let p: The plane is on time. Let q: The sky is clear. (a) p ˄ (¬ q) The plane is on time and the sky is not clear. (b) q → (p ˅ ¬ p) If the sky is clear, either the plane is on time or not. (c) p ↔ q The plane is on time if and only if the sky is clear. →˅¬↔˄ 2. Construct a truth table for each proposition. (a) [(p ˄ q) ˅ r] ↔ [(p ˄ r ) ˅ (q ˄ r)] p

q

r

p˄q

(p ˄ q) ˅ r

p˄r

q˄r

(p ˄ r) ˅ (q ˄ r )

[(p ˄ q) ˅ r] ↔ [(p ˄ r ) ˅ (q ˄ r)]

1 1 1 1 0 0 0 0

1 1 0 0 1 1 0 0

1 0 1 0 1 0 1 0

1 1 0 0 0 0 0 0

1 1 1 0 1 0 1 0

1 1 0 0 0 0 0 0

1 0 0 0 1 0 0 0

1 1 0 0 1 0 0 0

1 1 0 1 1 1 0 1

(b) [(p ˄ r) → (q ˄ → r)] → [(p ˄ q) ˅ r)] p

q

r

p˄r

¬r

1 1 1 1 0 0 0 0

1 1 0 0 1 1 0 0

1 0 1 0 1 0 1 0

1 0 1 0 0 0 0 0

0 1 0 1 0 1 0 1

q˄¬ r 0 1 0 0 0 1 0 0

(p ˄ r) → (q ˄ ¬ r)

p˄q

(p ˄ q) ˅ r

[(p ˄ r) → (q ˄ → r)] → [(p ˄ q) ˅ r)]

0 1 0 1 1 1 1 1

1 1 0 0 0 0 0 0

1 1 1 0 1 0 1 0

1 1 1 0 1 0 1 0

3. Prove the De Morgan’s Laws by constructing truth tables. (a) ¬ (p ˅ q) ↔ (¬ p) ˄ (¬ q) p

q

(p ˅ q)

1 1 0 0

1 0 1 0

1 1 1 0

¬ (p ˅ q) 0 0 0 1

¬p

¬q

(¬ p) ˄ (¬ q)

¬ (p ˅ q) ↔ (¬ p) ˄ (¬ q)

0 0 1 1

0 1 0 1

0 0 0 1

1 1 1 1

¬p

¬q

(¬ p) ˅ (¬ q)

¬ (p ˄ q) ↔ (¬ p) ˅ (¬ q)

0 0 1 1

0 1 0 1

0 1 1 1

1 1 1 1

(a) ¬ (p ˄ q) ↔ (¬ p) ˅ (¬ q) p

q

(p ˄ q)

1 1 0 0

1 0 1 0

1 0 0 0

¬ (p ˄ q) 0 1 1 1

4. Let U : = Letters in the English Alphabet {a, b, c, … x, y, z} A = {t, r, I, a, n, g, l, e, s} B = {s, q, u, a, r, e} C = {h, e, x, a, g, o, n, s} Determine the following: ∪∩ a) A ∪ (B ∩ C) ={t, r, i, a, n, g, l, e, s} b) (A ∪ B)’ ∩ C ={h, x, o} c) (A ∩ C) ∪ (B ∩ C) ={a, n, g, e, s} d) A ∩ (C ∩ U ) ={h, x, o} e) n [(A ∪ B) ∩ (B ∪ C)] =13 5. A survey of 90 customers were taken at Barnes & Noble regarding the types of books purchased. The survey found that 44 purchased mysteries, 33 purchased science fiction, 29 purchased romance novels, 13 purchased mysteries and science fiction, 5 purchased science fiction and romance novels, 11 purchased mysteries and romance novels, and 2 purchased all three types of books (mysteries, science fiction, romance novels). How many customer surveyed purchased (a) Mysteries only?

n(S ∪ R)’ ∩ n(M)= 22 (b) Mysteries and science fiction, but not romance novels? n(R)’ ∩ n(S ∩ M) = 11 (c) Mysteries or science fiction? n(S ∪ M) = 64 (d) Romance novels or mysteries, but not science fiction? n(S)’ ∩ n(R ∪ M)= 46 (e) Exactly two types (mysteries, science fiction, romance novels)? n(S ∩ M ∩ R)’ ∩ n(S ∩ M) ∪ n(S ∩ R) ∪ n(M ∩ R)= 25...