MAT2440
Take-home Practice 1
Instruction: Due time: in class on Tuesday, February 7, 2023.
Finish the following questions independently. No credit is given if no steps provided.
1. Translate the following English sentence to propositional logic.
Information: When translating from English sentences into logical form, "but" generally
means the same as "and".
a) You get an A in this class, but you do not do every exercise in textbook.
b) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries
are rip along the trail.
c) For you to get an A in this class, if and only if you have A in all your essay.
2. State the converse, inverse, and contrapositive of the following conditional statements.
Information: Identify the hypothesis and conclusion first. Rewrite the conditional statement in
form of hypothesis  conclusion. Then find the converse, inverses, and contrapositive.
a) If it snows tomorrow, I go shopping today.
b) You get a speeding ticket if you drive over 65 miles per hour.
3. Construct a truth table for the following compound proposition.
(𝑝𝑝 ∧ ¬𝑞𝑞) → (¬𝑝𝑝 ∨ 𝑟𝑟)
4. Use a truth table to verity the distributive law
𝑝𝑝 ∧ (𝑞𝑞 ∨ 𝑟𝑟) ≡ (𝑝𝑝 ∧ 𝑞𝑞) ∨ (𝑝𝑝 ∧ 𝑟𝑟)
5. Use De Morgan’s laws to find the negation of each of the following statements.
Steps: 1. Translate each statement to propositional logic. 2. Apply De Morgan’s laws to the
propositional logic. 3. Write an English statement according to the step 2.
Example: Jan is rich and happy.
Step 1: 𝑝𝑝 ∧ 𝑞𝑞 p: Jan is rich. q: Jan is happy.
Step 2: ¬(𝑝𝑝 ∧ 𝑞𝑞) ≡ ¬𝑝𝑝 ∨ ¬𝑞𝑞
Step 3: Jan is not rich or not happy.
a) Mei will run or walk to the gym tomorrow.
b) Jerry knows Jave but not C++.
6. Find the output of the combinatorial circuit.