Directions: Answer all questions completely according to all the directions given. In addition to your
book and notes, you are permitted to work with other students from the class, if you so desire. Direct
copying of answers is prohibited, and may result in a 0 grade for the assignment. When you are
finished, upload the homework as a MSWord (.docx) or Adobe (.pdf) document to the appropriate
assignment in Canvas. If you wish, you may print the homework and scan the completed pages. You
may also take photos of the pages and upload them, but be sure your answers are legible.
Section 1: Short Answer. (20 points)
1. Successfully performing a proof on an argument tells us what about that argument?
2. What two things must you indicate in the justification for every line in a proof?
3. Explain why the rule of implication “Addition” is valid.
4. Provide at least two differences between the rules of implication and the rules of
equivalence.
5.What is the Law of the Excluded Middle?
6.What is the Law of Non-contradiction?
7.Perform either version of Association on the following expression: M ⌵ [(N → ~N) ⌵ O]
8. Perform either version of DeMorgan’s Law on the following expression: M → ∼(∼N ⦁ O)
9. Perform either version of Material Equivalence on the following expression: M → ∼(∼N ↔ O)
10.Perform either version of Distribution on the following expression: (M → N) ⌵ (O ⦁ P)
Rules of Implication and Equivalence: Provide the symbolic form for the following valid inferences in the
appropriate space below. You may copy directly from your textbook. (10 Points)
Modus Ponens Modus Tollens Hypothetical Syllogism
Disjunctive Syllogism Constructive Dilemma Addition
Conjunction Simplification Double Negation
Commutation Association De Morgan’s Laws
Contraposition Exportation Distribution
Redundancy Material Equivalence Material Implication
Rules of Implication: Construct a formal proof of validity using ONLY the eight rules of implication as needed. (25
points)
1. A → (B ⌵ C)
2. D ⦁ A
3. ∴ B ⌵ C
1. P ⦁ (Q → ~R)
2. P ⌵ S
3. ~(Q → ~R)
4. ∴ S
1. (P ⦁ Q) → (S ↔ T)
2. (~R ⌵ ~S) ⦁ ~~S
3. ~R → P
4. Q ⌵ ~S
5. ∴ S ↔ T
1. (A ⌵ Q) → [~B → (C ⌵ ~D)]
2. ~B ⦁ A
3. B ⌵ ~~D
4. ∴ C
1. [W ⌵ (Y ⌵ ~W)] ⦁ ~X
2. Z → X
3. {(W → V) ⦁ [(Y ⌵ ~W) → ~T]} ⌵ Z
4. ∴ V ⌵ ~T
Rules of Implication and Equivalence: Construct a formal proof of validity using any of the eight rules of
implication or ten rules of equivalence as needed. If you use additional paper, be sure to indicate your name on
each sheet. (45 pts)
1. M ⌵ ~Q
2. N → ~M
3. Q
4. ∴ ~N
1. Q
2. ∴ P → Q
1. P ↔ Q
2. ~Q
3. ∴ ~P
1. (M ⦁ L) ↔ T
2. ∴ M → (L → T)
1. ~(P ⦁ Q)
2. ~P → R
3. ~Q → S
4. ~S ⦁ T
5. ∴ R
1. P ⌵ (Q ⌵ R)
2. ~R ⦁ ~Q
3. S ↔ (P ⌵ P)
4. ∴ S
1. ~{[N → (M → O)] ⌵ P}
2. ~[~(M → O) → ~N] → Q
3. ∴ Q
1. (T ⌵ P) ↔ (M ⌵ N)
2. ~[~(T ⌵ P) ⦁ ~(M ⌵ N)]
3. ~N
4. ∴ M
1. P ⦁ (~T ⦁ B)
2. B ↔ (R → T)
3. R ⌵ (P → Q)
4. ∴ Q