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Tautology in Math: Definition & Examples

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  • 0:01 What Is a Tautology?
  • 0:23 Truth Table
  • 1:28 Additional Examples
  • 2:22 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.

Discover what a tautology is, and learn how to determine if a statement is a tautology by constructing a truth table. Test your skills with a short quiz.

What Is a Tautology?

A tautology is a statement that is always true, no matter what. If you construct a truth table for a statement and all of the column values for the statement are true (T), then the statement is a tautology because it's always true! The famous saying 'I cannot tell a lie' may come to mind when studying tautologies.

Truth Table

The statement 'I will either get paid or not get paid' is a tautology since it is always true. Most of the time the logic statements or arguments that we are trying to analyze are more complicated than this, or we are only given the symbolic representation of the statement and not the statement itself.

If you are given a statement and want to determine if it is a tautology, then all you need to do is construct a truth table for the statement and look at the truth values in the final column. If all of the values are T (for true), then the statement is a tautology.

Let's look at an example.

Remembering our statement 'I will either get paid or not get paid', we can use p to represent the statement 'I will get paid' and not p (written ¬p) to represent 'I will not get paid.'

p: I will get paid

¬p: I will not get paid

So, p V ¬p: I will either get paid or not get paid

A truth table for the statement would look like:

Negation Tautology Truth Table

Looking at the final column in the truth table, you can see that all the truth values are T (for true). Whenever all of the truth values in the final column are true, the statement is a tautology. So, our statement 'I will either get paid or not get paid' is always a true statement, a tautology.

Additional Examples

Let's look at another example. To determine if the argument p -> (p V q) is a tautology or not, we can construct a truth table and check the truth values in the final column. Here's the truth table:

Truth Table for Implication Tautology

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