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# COMPLETING A TRUTH TABLE

A truth table involving n statements has rows. So a table for a statement involving p and q has 4 rows and a table involving p, q and r has 8 rows etc. Here we give an example of completing a truth table with 8 rows. Typically, the table is set up as follows. We hold the first two statements true and consider the two cases of the last statement being true or false giving the first two rows of the table. Next we let the second statement be false and again consider both possible truth values of the last statement (giving the first 4 rows of the table). Finally, we let the first statement be false and consider the possible truth values of the last two statements, first with the second statement true and the last statement taking on both possible values and then the second statement being false and the last statement taking both values. We complete the truth table for the statement will have eight rows.

As in algebra, we find the truth values of parenthesized expressions first. So to complete this example we'll first find the truth value of . Second, we'll find the truth value of and finally find the truth value of the complete statement.

We look up the truth values for the conditional involving two statements to find the truth values of . These values are now entered in the table. Note that it is important to know the elementary truth tables. Also, it is probably better to memorize in the form ``T T is T, T F is F'' etc., rather than think in terms of p and q.

Note that whenever p is false that ? is true. In fact, it is only false when the first statement is true and the second is false. Next we fill in the truth values of .

We look up the truth values for the conjunction involving two statements to find the truth values of . These values are now entered in the table. Again, think of the conjunction of two statements is T except when both statements are false.

Finally, we use the truth values we have entered to fill in the truth values of the complete statement

We look up the truth values for the conjunction involving two statements to find the truth values of . These values are now entered in the table.

Finally, we use the truth values we have entered to fill in the truth values of the complete statement.

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Peter Williams
Sun Oct 13 00:22:23 PDT 1996