Discrete Structures, Logic, and Computability, 4th Edition

Chapter 6

Section 6.2

1. a. $((\neg P)\wedge Q)\to (P\vee R)\mathrm{.}$

c. $(A\to (B\vee \left(\right((\neg \mathrm{\hspace{0.17em}}C)\wedge D)\wedge E)\left)\right)\to F\mathrm{.}$

2. a. $(P\vee Q\to \neg \mathrm{\hspace{0.17em}}R)\vee \neg \mathrm{\hspace{0.17em}}Q\wedge R\wedge P\mathrm{.}$

3. $(A\to B)\wedge (\neg \mathrm{\hspace{0.17em}}A\to C)\text{or}(A\wedge B)\vee (\neg \mathrm{\hspace{0.17em}}A\wedge C)\mathrm{.}$

5. $A\wedge \neg \mathrm{\hspace{0.17em}}B\to \text{False}\equiv \neg \mathrm{\hspace{0.17em}}(A\wedge \neg \mathrm{\hspace{0.17em}}B)\vee \text{False}\equiv \neg \mathrm{\hspace{0.17em}}(A\wedge \neg \mathrm{\hspace{0.17em}}B)\equiv \neg \mathrm{\hspace{0.17em}}A\vee \neg \neg \mathrm{\hspace{0.17em}}B\equiv \neg \mathrm{\hspace{0.17em}}A\vee B\equiv A\to \mathrm{B.}$

7. a. If B = True, then the wff is true. If B = False and A = True, then the wff is false.

c. If A = True, then the wff is true. If A = False and C = True, then the wff is false.

e. If B = True, then the wff is true. If B = False and A = C = True, then the wff is false.

8. a. If C = True, A → C is true, so the wff is trivially true too. If C = False, then the wff becomes (A → B) ∧ (B → False) → (A → False), which is equivalent to (A → B) ∧ ¬ B → ¬ A. If A = False, then the wff is trivially true. If A = True, the wff becomes

(True → B) ∧ ¬ B → False ≡ ...