Practice Work

1.1

Show $(p\wedge q)\to r$ and $(p\to r)\wedge (q\to r)$ are not logically equivalent.

$p$	$q$	$r$	$p\wedge q$	$(p\wedge q)\to r$	$p\to r$	$q\to r$	$(p\to r)\wedge (q\to r)$	$((p\wedge q)\to r)⟺((p\to r)\wedge (q\to r))$
T	T	T	T	T	T	T	T	T
T	T	F	T	F	F	F	F	T
T	F	T	F	T	T	T	T	T
T	F	F	F	T	F	T	F	F
F	T	T	F	T	T	T	T	T
F	T	F	F	T	T	F	F	F
F	F	T	F	T	T	T	T	T
F	F	F	F	T	T	T	T	T

Hence they are not logically equivalent since they produce different results when TFF or FTF are assigned to p,q,r respectively.

1.2

Formalise the following statements in propositional logic then:

• decide whether it is consistent or not
• explain answer

If Bill is at home, the dog is happy.
There is a thunderstorm and the dog is not happy.
Bill is at home.

Let $h$ be Bill is at home. Let $d$ be the dog is happy. Let $t$ be that there is a thunderstorm.

Hence,

• $h\to d$
• $t\wedge \neg d=\text{T}$
• $h=\text{T}$

These statements are not consistent because since Bill is at home $(h=\text{T})$, the dog must be happy $(d=\text{T})$ but this contradicts that the dog is not happy $(t\wedge \neg d=\text{T})$.

1.3

Formalise the following argument in propositional logic and decide whether it is logically correct:

$$\begin{array}{rl}& \underline{\begin{array}{l}\text{If it is raining and the birds are singing, then Joe is not happy.}\\ \text{It is not a good day and it is raining.}\\ \text{If the birds are singing and Joe is happy, then it is a good day.}\end{array}}\\ \therefore & \text{If Joe is not happy, then the birds are not singing.}\end{array}$$

Let $r$ be that it is raining. Let $b$ be that the birds are singing. Let $h$ be that Joe is happy. Let $d$ be that it is a good day.

Hence,

• $p_1=(r\wedge b)\to \neg h$
• $p_2=(\neg d\wedge r)$
• $p_3=(b\wedge h)\to d$
• $q=\neg h\to \neg b$

We know that it is not a good day and it is raining $(\neg d\wedge r)=\text{T}$, so:

• $d$ is false
• $r$ is true.

Going back to the formalisation:

$$\begin{array}{rl}(p_1\wedge p_2\wedge p_3)& \to q\\ ((r\wedge b)\to \neg h)\wedge (\neg d\wedge r)\wedge ((b\wedge h)\to d)& \to (\neg h\to \neg b)\\ ((T\wedge b)\to \neg h)\wedge (\neg F\wedge T)\wedge ((b\wedge h)\to F)& \to (\neg h\to \neg b)\\ (b\to \neg h)\wedge (\neg (b\wedge h))& \to (\neg h\to \neg b)\end{array}$$

If the statements are logically correct, then $(p_1\wedge p_2\wedge p_3)\to q$ is a tautology.

$b$	$h$	$\neg b$	$\neg h$	$(\neg h)\to (\neg b)$	$b\wedge h$	$\neg (b\wedge h)$	$b\to (\neg h)$	$(b\to (\neg h))\wedge (\neg (b\wedge h))$	$((b\to (\neg h))\wedge (\neg (b\wedge h)))\to ((\neg h)\to (\neg b))$
F	F	T	T	T	F	T	T	T	T
F	T	T	F	T	F	T	T	T	T
T	F	F	T	F	F	T	T	T	F
T	T	F	F	T	T	F	F	F	T

We have just proved that $(p_1\wedge p_2\wedge p_3)\to q$ is not a tautology hence the statements are not logically correct.