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High School Geometry: Help and Review13 chapters | 162 lessons

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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.

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.

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:

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.

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:

We can see from looking at the final column (the one on the far right), that all of the truth values are T (for true), so the statement *p* -> (*p* V *q*) is a tautology. We do not even need to know what the statements *p* and *q* mean to determine that the argument is always true! If any of the truth values in the final column are false (F), then the statement (or argument) is not a tautology.

Okay, one last example.

Let's determine if the argument (*p* -> *q*) V (*q* -> *p*) is a tautology. The truth table looks like this:

We can see that this is another tautology since all of the truth values in the final column are true. Remember, if any of these values are false, the statement would not be a tautology.

A **tautology** is a statement that is always true. If you are given any statement or argument, you can determine if it is a tautology by constructing a truth table for the statement and looking at the final column in the truth table. If all of the truth values in the final column are true, then the statement is a tautology. If any of the truth values in the final column are false, then the statement is not a tautology.

This lesson prepares students to:

- Define tautology
- Construct a truth table
- Interpret a truth table to determine if a statement is a tautology

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High School Geometry: Help and Review13 chapters | 162 lessons

- Critical Thinking and Logic in Mathematics 4:27
- Logical Fallacies: Hasty Generalization, Circular Reasoning, False Cause & Limited Choice 4:47
- Logical Fallacies: Appeals to Ignorance, Emotion or Popularity 8:53
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- Logical Math Connectors: Conjunctions and Disjunctions 3:39
- Conditional Statements in Math 4:54
- Logic Laws: Converse, Inverse, Contrapositive & Counterexample 7:09
- Direct Proofs: Definition and Applications 7:11
- Geometric Proofs: Definition and Format 8:35
- Basis Point: Definition, Value & Conversion
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