Learn what truth tables are and what they are used for in logic. Discover the basic rules behind constructing truth tables and explore the concepts of negation, conjunction, disjunction, and implication.

## Definition of a Truth Table

A **truth table** is a mathematical table used to determine if a compound statement is true or false. In a truth table, each statement is typically represented by a letter or variable, like *p*, *q*, or *r*, and each statement also has its own corresponding column in the truth table that lists all of the possible truth values.

In this lesson, we will learn the basic rules needed to construct a truth table and look at some examples of truth tables.We may not sketch out a truth table in our everyday lives, but we still use the logical reasoning that truth tables are built from to evaluate whether statements are true or false. Let’s say we are told ‘If it is raining outside, then the football game is cancelled.’ We can use logical reasoning rules to evaluate if the statement is true or false and maybe make some backup plans! Let’s check out some of the basic truth table rules.

## Input Values

Let’s take the statement, ‘It is raining outside.

‘ This statement, which we can represent with the variable *p*, is either true or false.*p* = It is raining outsideIf it is raining, then *p* is true. If it isn’t raining, then *p* is false.The **negation** of a statement, called *not p*, is the statement that contradicts *p* and has the opposite truth value.*not p* = It is not raining outsideIf it is raining outside, then *not p* is false. If it isn’t raining outside, then *not p* is true.Here is how both of these possibilities are represented in a truth table in which *T* represents true, and *F* represents false:

## Implication

An **implication** is a conditional ‘if-then’ statement like ‘If it is raining outside, then the football game is cancelled.’ Implications can seem tricky at first since they are only false when the antecedent (the ‘if’ part) is true, and the consequent (the ‘then’ part) is false.So, the implication ‘If it is raining outside, then the football game is cancelled.’ will only be false if *p* is true and *q* is false.

In other words, it is raining outside, but the football game is not cancelled. The implication is false because the promise of the implication was broken. However, if it is not raining (*p* is false), then the promise of the implication cannot be broken since the first part (the ‘if’ part) never happened, so the implication holds true.

The implication does not say what happens if it is not raining outside!Here is what the implication truth table looks like:

## Constructing Truth Tables

Now that you’ve seen some of the basic truth tables, you can start constructing your own to evaluate more complicated compound statements. It helps to have some tips to make the tables in an organized way so you don’t leave out any possibilities.

**Step 1:** Count how many statements you have, and make a column for each statement.**Step 2:** Fill in the different possible truth values for each column. If there is only one statement, then the first column will only have two cases (TF). If there are two statements, then there are four different possible cases: the first column will be (TTFF) and the second will alternate (TFTF). If there are three statements, then there are eight different possible cases: the first column will be (TTTTFFFF), the second will be (TTFFTTFF), and the third will alternate (TFTFTFTF).

**Step 3:** Add a column for each negated statement, and fill in the truth values.**Step 4:** Add columns for any conjunctions, disjunctions, or implications that are inside of parentheses or any grouping symbols.**Step 5:** Add a final column for the complete compound statement.

## Contrapositive Example

Using the two statements from before, let’s construct a truth table for the compound statement, ‘If the football game is not cancelled, then it is not raining outside.’ This is the contrapositive of the original implication.Recall:*p* = It is raining outside*q* = The football game is cancelledWe can write the contrapositive as *not q* then *not p*.Step 1: We have two statements (*p* and *q*), so we need two columns.Step 2: Since there are two statements, we will have four different cases. The first column will be (TTFF), and the second column will be (TFTF).

Step 3: Add two columns: one for *not p* and one for *not q*.Step 4: Add the final column for *not q* then *not p*.

## Lesson Summary

We can use a **truth table** as an organized way of seeing all of the possibilities when evaluating if a compound statement is true or false. A letter or variable typically represents statements.

To find out if a statement is true or false, we use logical reasoning rules, such as **negation**, **conjunction**, **disjunction**, and **implication**. These rules can also be used to construct columns in a truth table, which typically includes two case columns for each statement and separate columns for each negated statement and the complete compound statement.