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.

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