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Truth Tables

A convenient and helpful way to organize truth values of various statements is in a truth table. A truth table is a table whose columns are statements, and whose rows are possible scenarios. The table contains every possible scenario and the truth values that would occur. One of the simplest truth tables records the truth values for a statement and its negation.

Figure %: The truth table for p , âàüp
Remember that a statement and its negation, by definition, always have opposite truth values. This is shown in the truth table.

Truth tables get a little more complicated when conjunctions and disjunctions of statements are included. Below is the truth table for p , q , pâàçq , pâàèq . Notice that all the values are correct, and all possibilities are accounted for.

Figure %: The truth table for p , q , pâàçq , pâàèq

The truth table for an implication, or conditional statement looks like this:

Figure %: The truth table for p , q , pâá’q
The first two possibilities make sense. If p is true and q is true, then (pâá’q) is true. Also, if p is true and q is false, then (pâá’q) must be false. The last two possibilities, in which p is false, are harder to decide upon. If p is false, then the implication with p as the hypothesis will not meet its condition (that p be true) so q does not have to be either true or false. Either way, the implication has not been denied, because its condition was not met, so the implication stands as true.

Now that the truth table for a standard conditional statement is understood, we'll take a look at the truth table for its inverse, converse, and contrapositive.

Figure %: The truth table for an implication and its inverse, converse, and contrapositive
Notice that the contrapositive has the same truth values as the original implication. Also notice that the converse and the inverse are each other's contrapositive, and therefore have equivalent truth values.

One more thing should be said of truth tables: they can hold more than two different statements. You could have p , q , r , s , and t in the same truth table. There would then be 32 possible scenarios ( 25 ), so the table would have 5 columns and 32 rows.

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