Tutorial list 1

1

Let $P$ be that it is below freezing. Let $Q$ be that it is snowing.

(a)

1. $P\wedge Q$
2. $P\wedge \neg Q$
3. $P\vee Q$
4. $P\to \neg Q$
5. $\cancel{P⟺Q}Q\to P$ (one implies the other)
6. $P\to Q$
7. $Q⟺P$
8. $P\to Q$

(b)

1. It is not below freezing.
2. If it is below freezing then it is snowing.
3. It is below freezing or it is snowing or both.
4. It is below freezing and it is snowing.
5. If it is not below freezing then it is not snowing.
6. It is below freezing if and only if it is snowing.
7. It is not below freezing and it is not snowing.
8. It is not below freezing or it is below freezing and snowing or both. (both would be a contradictory statement)

title:
Correct answers.

2

Let $u$ be that the system software is being upgraded. Let $f$ be that users can access the file system. Let $s$ be that users can save new files.

Hence:

• $u\to \neg f$
• $f\to s$
• $\neg s\to \neg u$

title:
Correct answer.

3

Construct syntactic decomposition tree of:

$$(\neg P\to \neg Q\vee R)\wedge \neg (\neg Q⟺\neg R)$$

*Done on paper.

4

Consider $(P\vee \neg Q)\to \neg (Q\vee \neg P)$.

• Construct a truth table.

interpretation	$p$	$q$	$\neg p$	$q\vee (\neg p)$	$\neg (q\vee (\neg p))$	$\neg q$	$p\vee (\neg q)$	$(p\vee (\neg q))\to (\neg (q\vee (\neg p)))$
$v_0$	0	0	1	1	0	1	1	0
$v_1$	0	1	1	1	0	0	0	1
$v_2$	1	0	0	0	1	1	1	1
$v_3$	1	1	0	1	0	0	1	0

It is a contingency as it is neither:

• a tautology as not all values are true
• a contradiction as some values are true

title:
Correct answer.

5

1. If there are winds from the south, then there is a spring thaw.
2. If you bought the computer less than a year ago, then the warranty is good.
3. If Lenny cheats then Lenny gets caught.
4. If you pay a subscription fee then you can access the website. If you can access the website, then you have paid a subscription fee.
5. If you have a valid password then you can log on to the server. If you can log on to the server then you must have a valid password.
6. If the water is not too cold then Jan will go swimming.
7. If you learn discrete mathematics then you will find a good job.

6

$$\begin{array}{rl}\neg P_4\to \neg P_2\oplus P_3\oplus P_1& =\neg F\to \neg T\oplus F\oplus T\\ & =T\to (F\oplus F)\oplus T\\ & =T\to T\\ & =T\end{array}$$

title:
Should be using $\lor$ not $\oplus$.
 
$$
	\begin{aligned}
		\neg P_4 \rightarrow \neg P_2 \lor P_3 \lor P_1
			&= \neg F \rightarrow \neg T \lor F \lor T \\
  		&= T \rightarrow (F \lor F) \lor T \\
  		&= T \rightarrow T \\
  		&= T
	\end{aligned}
$$
 
Find error above, not sure.

7

interpretation	$p$	$q$	$p\Vert q$
$v_0$	0	0	1
$v_1$	0	1	1
$v_2$	1	0	1
$v_3$	1	1	0

interpretation	$p$	$q$	$p\wedge q$	$\neg (p\wedge q)$
$v_0$	0	0	0	1
$v_1$	0	1	0	1
$v_2$	1	0	0	1
$v_3$	1	1	1	0

Hence they are logically equivalent.