## Number Systems: Problem Sheet 6

Here are some hints for Problem Sheet 6 given to people in my office hour on Wednesday.

Question 2. Two statements are logically equivalent if and only if they correspond to equal columns on a truth table. For instance, too show that $A \Rightarrow B$ and $\neg A \vee B$ are logically equivalent, make a truth table with columns $A$, $B$, $A \Rightarrow B$ and $\neg A \vee B$, and check that the final two columns are equal.

Question 3. For (b), you should find that if $z=2i$ then $r=2$ and $\sin \theta = 1$, $\cos \theta = 0$. Now draw the graph of $\sin$ and argue that $\theta = \pi/2 + 2n\pi$ for some $n \in \mathbb{Z}$. Please try to use implies signs ($\Rightarrow$) to show the logical structure of your argument.

For (c): I suggest you start by converting the statement ‘if $r=2$ and $\theta = \pi/2 + 2n \pi$ for some $n \in \mathbb{Z}$ then $z =2i$‘ into symbols. This statement is of the form ‘if … then …’, so we need to use $\Rightarrow$. Use $(\exists n \in \mathbb{Z}) (\theta = \pi/2 + 2n\pi)$ to express the condition on $\theta$ and $\wedge$ (conjunction) to put the condition on $r$ and $\theta$ together. Finally you will need to specify that the converse implication $\Longleftarrow$ also holds. For this $\Longleftrightarrow$ is ideal.

Question 4. You can do this with three eight-rowed truth tables. A statement is a tautology if and only if it corresponds to a column of a truth table where the only entry is $T$. It might be good to start off with an easier tautology, for example $P \implies (Q \implies P)$ can be checked using a four-row truth table.

For a direct argument, it is very useful to notice that $A \Longrightarrow B$ is false if and only if $A$ is true and $B$ is false. So if (a) is false then $P \Longrightarrow Q$ and $Q \Longrightarrow R$ must be true and $P \Longrightarrow R$ must be false. Hence $P$ must be true and $R$ false, but then since $P \Longrightarrow Q$ is true, $Q$ is true, and since $Q \Longrightarrow R$ is true, $R$ must be true. So $R$ is true and false, a contradiction.

Question 6. For example, to get the truth table where the third column is (from the top) $F$, $F$, $T$, $F$, you could take $\neg A \wedge B$. You can also use just $A$ and $B$ on their own: there is no requirement that you use both propositions.