Question

6. Show that ~ (p → q) and p ∧~q are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the ~(p → q) is p ∧~q.

7.Let p and q be the propositions.
p: I bought a lottery ticket this week.
q: I won the million-dollar jackpot on Friday.
a) Form a tautology using p. Express the tautology in English sentence.
b) Form a tautology using q. Express the tautology in English sentence.
c) Form a contradiction using p. Express the contradiction in English sentence.
d) Form a contradiction using q. Express the contradiction in English sentence.

8. If you have a tautology r and you negate r, what kind of sentence do you get?
a. A tautology
b. A contradiction
c. A sentence that is neither a contradiction nor a tautology
d. You can’t tell—it could be any of (a), (b), or (c).

Answer 1

Answer on Question #72465 – Math – Discrete Mathematics

6. Show that $\sim (\mathrm{p} \rightarrow \mathrm{q})$ and $\mathrm{p} \land \sim \mathrm{q}$ are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the $\sim (\mathrm{p} \rightarrow \mathrm{q})$ is $\mathrm{p} \land \sim \mathrm{q}$.)

$$\begin{array}{l} \sim (\mathrm{p} \rightarrow \mathrm{q}) \quad \mathrm{p} \land \sim \mathrm{q} \\ \begin{array}{ccccccc} \mathbf{0} & 1 & 1 & 1 & 1 & \mathbf{0} & 0 & 1 \\ \mathbf{1} & 1 & 0 & 0 & 1 & \mathbf{1} & 1 & 0 \\ \mathbf{0} & 0 & 1 & 1 & 0 & \mathbf{0} & 0 & 1 \\ \mathbf{0} & 0 & 1 & 0 & 0 & \mathbf{0} & 1 & 0. \end{array} \\ \end{array}$$

So, both formulas have the same truth table, then they're logically equivalent.

7. Let $\mathrm{p}$ and $\mathrm{q}$ be the propositions.

p: I bought a lottery ticket this week.

q: I won the million-dollar jackpot on Friday.

a) Form a tautology using p. Express the tautology in English sentence.

p ∨ ¬p: I bought a lottery ticket this week or I didn’t buy a lottery ticket this week.

b) Form a tautology using q. Express the tautology in English sentence.

q ∨ ¬q: I won the million-dollar jackpot on Friday or I didn’t win the million-dollar jackpot on Friday.

c) Form a contradiction using p. Express the contradiction in English sentence.

p ∧ ¬p: I bought a lottery ticket this week and I didn’t buy a lottery ticket this week.

d) Form a contradiction using q. Express the contradiction in English sentence.

q ∧ ¬q: I won the million-dollar jackpot on Friday and I didn’t win the million-dollar jackpot on Friday.

8. If you have a tautology r and you negate r, what kind of sentence do you get?

b. A contradiction.

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