Discrete Mathematics Answers

1. Let R(x, y, z): x² + y² = z². Find the truth values of the propositions R(3, 4, 5) and R(2, 2, 4).

2. Let P(x, y): x = y+1. Find the truth values of the propositions P(1, 3) and P(2, 1).

Construct a truth table for each of these compound propo￾sitions.

a) p → ¬p b) p ↔ ¬p

c) p ⊕ (p ∨ q) d) (p ∧ q) → (p ∨ q)

e) (q → ¬p) ↔ (p ↔ q)

f ) (p ↔ q) ⊕ (p ↔ ¬q)

Let P(x,y) : x is a factor of y 

for numbers 3 & 4, use this domains: x = { 2, 6, 8, 9 }, y = { 11, 13, 17, 81 }

Find the following then identify their truth values.

1. P (1, 10)
2. P (3, 5)
3. ∀x P(x, y)
4. ∃x P(x, y)

Suppose that a statement of the form ∀xP(x) is false. How can this be proved?

Show how bitwise operations on bit strings can be

used to fifind these combinations of A = {a, b, c, d, e},

B

= {b, c, d, g, p, t, v}, C = {c, e, i, o, u, x, y, z}, and

D

= {d, e, h, i, n, o, t, u, x, y}.

a)

A ∪ B

b)

A ∩ B

c)

(A ∪ D) ∩ (B ∪ C)

d)

A ∪ B ∪ C ∪ D

B. Write each statement into its symbolic form.(3 pts each)

Let

x: PJ is a mathematician

y: MJ is a programmer

a. PJ is not a mathematician.

b. PJ is a mathematician while MJ is a programmer.

c. If PJ is a mathematician then MJ is not a programmer.

d. PJ is a mathematician or if PJ is a mathematician then MJ is a

programmer.

e. Either PJ is a mathematician and MJ is a programmer, or neither PJ is

a mathematician nor MJ is a programmer.

C. Show whether or not p → q ≡ (p ^ q) v (𝒑

̅ ^ 𝒒

̅) (10 pts)