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 and are logically equivalent, make a truth table with columns , , and , and check that the final two columns are equal.
Question 3. For (b), you should find that if then and , . Now draw the graph of and argue that for some . Please try to use implies signs () to show the logical structure of your argument.
For (c): I suggest you start by converting the statement ‘if and for some then ‘ into symbols. This statement is of the form ‘if … then …’, so we need to use . Use to express the condition on and (conjunction) to put the condition on and together. Finally you will need to specify that the converse implication also holds. For this 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 . It might be good to start off with an easier tautology, for example can be checked using a four-row truth table.
For a direct argument, it is very useful to notice that is false if and only if is true and is false. So if (a) is false then and must be true and must be false. Hence must be true and false, but then since is true, is true, and since is true, must be true. So is true and false, a contradiction.
Question 6. For example, to get the truth table where the third column is (from the top) , , , , you could take . You can also use just and on their own: there is no requirement that you use both propositions.