## Content

### More on truth tables

In this section we concentrate on truth tables for two ways of combining propositions. These ways are important in certain searches, for example on the web.

The first method of combination is when we are looking for both a proposition $p$ and a proposition $q$. So we want both $p$ and $q$. We'll represent this as the proposition $pq$ and show its truth table in Table 6. You can see from the table, perhaps, why we write the combine proposition as $pq$.

#### Table 6: The truth table for $pq$

$p$ $q$ $pq$
0 0 0
0 1 0
1 0 0
1 1 1

As you would expect, the result for $pq$ is only 1 (true) when both $p$ and $q$ are true.

You might want to search for female politicians so $p$ might be 'the person is a woman' and $q$ 'the person is a member of parliament'. Clearly you will get quite a different, and perhaps less useful result, if you searched for 'either the person is a woman or the person is a member of parliament'. But you might want to search for flights that 'go from Melbourne to London either through Singapore or Hong Kong'. So the combination either $p$ or $q$ is worth considering. This proposition is represented by $p + q$. We show the truth table of this combination in Table 7.

#### Table 7: The truth table for $p+q$

$p$ $q$ $p+q$
0 0 0
0 1 1
1 0 1
1 1 1

This table has only one false entry and that is for flights that go neither of the ways stipulated.

Questions

1. Which of the following propositions are of the form $pq$ and which of the form $p + q$? What are $p$ and $q$ in each case?
1. $\hspace{9px}$May does the washing and Ria does the drying;
2. $\hspace{5px}$Fred watches sport or detective programmes;
3. Today it will be 25$^{\circ}$ and there will be showers;
4. $\hspace{2px}$Marilyn will wear a skirt or a pair of jeans;
5. $\hspace{6px}$Leanne will wear a skirt with jeans underneath;
6. $\hspace{2px}$Spot is a black dog.

Questions

1. Suppose that $p$ is the proposition 'Alex swam at the Olympics' and $q$ is 'Alex won a gold medal'. What do the following propositions mean in words?
1. $\hspace{9px}pq;$
2. $\hspace{5px}p + q;$
3. $pq';$
4. $\hspace{2px}p + q';$
5. $\hspace{6px}p + p'q;$
6. $\hspace{2px}p'q + pq'.$
2. Write truth tables for the compound propositions (iii) to (vi) of Q48. What is interesting about (v)?

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