Discrete Mathematics Answers

X=factors of 30 R=x/y draw hasse diagram

Given the following:

• g: "You can graduate."
• m: "You owe money to the college."
• r: "You have completed the requirements of your major."
• b: "You have an overdue book."

Translate "You can graduate only if you have completed the requirements of your major, you do not owe money to the college, and you do not have an overdue book." into a propositional logic.

Find the graph of the function f(n)=1-n from integers to integer

a) Find simpler statement forms that are logically equivalent to 𝑝 ⊕ 𝑝 and (𝑝 ⊕ 𝑝) ⊕ 𝑝.

b) Is (𝑝 ⊕ 𝑞) ⊕ 𝑟 ≡ 𝑝 ⊕ (𝑞 ⊕ 𝑟)? Justify your answer.

c) Is (𝑝 ⊕ 𝑞) ∧ 𝑟 ≡ (𝑝 ∧ 𝑟) ⊕ (𝑞 ∧ 𝑟)? Justify your answer. 

If 𝑝 → 𝑞 is false, can you determine the truth value of (~𝑝)˅(𝑝 ↔ 𝑞)? Explain your answer.

If 𝑝 → 𝑞 is true, can you determine the truth value of ~(𝑝 → 𝑞)˄~𝑝? Explain your answer. 

1. Given the following:

• g: "You can graduate."
• m: "You owe money to the college."
• r: "You have completed the requirements of your major."
• b: "You have an overdue book."

Translate "You can graduate only if you have completed the requirements of your major, you do not owe money to the college, and you do not have an overdue book." into a propositional logic.

2. Show that  are logically equivalent. 

3. Show, by the use of the truth table (truth matrix), that the  is a tautology.

A. List the members of the following sets

1. {x| x is real numbers and x2 = 1}

2. {x| x is an integer and -4 < x ≤ 3}

B. Use set builder notation to give description of each of these sets.

1. {a, e,i ,o, u}

2. {=2, -1, 0, 1, 2}

C. Let A= (a, b, c), B = (x, y) and C = (0, 1)

Find:

1. A U C

2. C x B

3. B – A

4. (A ∩ C) U B

Use the properties to verify the logical equivalences in the following. Supply a reason for each

step.

a. (p ∧∼ q) ∨ p ≡ p

b. p ∧ (∼ q ∨ p) ≡ p

c. ∼ (p ∨∼ q) ∨ (∼ p ∧∼ q) ≡ ∼ p

d. ∼ ((∼ p ∧ q) ∨ (∼ p ∧∼ q)) ∨ (p ∧ q) ≡ p

e. (p ∧ (∼ (∼ p ∨ q))) ∨ (p ∧ q) ≡ p

Prove the equivalence of the following in three different ways (truth table, simplification,

each is a logical consequence of the other): p → (q ∨ r) ≡ (p ∧ ~q) → r.

If p → q is false, can you determine the truth value of (~p)˅(p ↔ q)? Explain your answer.

If p → q is true, can you determine the truth value of ~(p → q)˄~p? Explain your answer.