Truth Table: Definition, Rules & Examples

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  • 0:02 Definition of a Truth Table
  • 0:52 Input Values & Conjunction
  • 2:08 Disjunction & Implication
  • 3:28 Constructing Truth Tables
  • 4:41 Contrapositive Example
  • 5:30 Lesson Summary
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Lesson Transcript
Instructor: Julie Crenshaw

Julie has a Master's Degree in Math Education with a Community College Teaching Emphasis, and has been teaching college mathematics for over 10 years.

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 outside

If 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 outside

If 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:

Negation Truth Table


A conjunction is a compound statement representing the word 'and.' For example, we have the following two statements:

p = It is raining outside
q = The football game is cancelled

The conjunction of p and q is 'It is raining outside, and the football game is cancelled.' This statement will only be true if both p and q are true; that is, if it is raining outside and the football game is cancelled. If either p or q is false, then the conjunction is false.

Here is the truth table showing the possibilities of a conjunction:

Conjunction Truth Table


A disjunction is a compound statement representing the word 'or.' In order for a disjunction to be true, one or both of the original statements has to be true. The disjunction of the above statements p or q is 'It is raining outside, or the football game is cancelled.' This statement is true if p or q or both statements are true.

Here is what the truth table looks like:

Disjunction Truth Table


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:

Implication Truth Table

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