A tautology in math (and logic) is a compound statement (premise and conclusion) that always produces truth. No matter what the individual parts are, the result is a true statement; a **tautology** is always true. The opposite of a tautology is a **contradiction** or a **fallacy**, which is "always false".

Tautologies are typically found in the branch of mathematics called **logic**. They use their own special symbols:

- $\mathbf{\wedge}$ means $AND$
- $\mathbf{=}$ signifies $\u201cisequivalentto\u201d$
- $\mathbf{\neg}$ indicates $NEGATION$
- $\mathbf{\sim}$ shows $NOT$
- $\mathbf{\vee}$ means $OR$
- $\mathbf{\to}$ signifies $\u201cimplies\u201d$ or $\u201cif-then\u201d$

$p$, $~p$ and $q$ all signify the statements, with $p$ generally reserved for the first one and either $~p$ or $q$ for the second statement

You can "translate" tautologies from ordinary language into mathematical expressions. To translate the compound statement, "I will give you $5 or I will not give you $5," we could write:

$p\vee ~p$

The two statements match the two parts, with the connector symbolized by $\mathbf{\vee}$:

$p$ takes the place of "I will give you $5"

$\mathbf{\vee}$ means the word "or"

$~p$ takes the place of "I will not give you $5"

We can determine the two conditions of this statement (either I give you $5 or I don't) and see that both produce valid answers:

- I give you $5, so the first statement is true and the second is false, producing a true statement.
- I do not give you $5, so the first statement is false and the second is true. This again produces a true statement.

Constructing a **truth table** helps make the definition of a tautology more clear. A truth table tests the various parts of any logic statement, including compound statements.

The first part of the compound statement, the premise, is symbolized in the first column. Logical **connectors** (words that tie the two statements together) are words like or, and, if. They provide conditions like sequence, reason and purpose, opposition and/or unexpected result, and so forth.

The conclusion or second statement, following the logical connector, is symbolized in the second column. The third column of the truth table shows the relationship between the two statements as either true, $T$, or false, $F$.

If every result in the third column is $T$, True, then the compound statement is a tautology. Here is a simple truth table built from the compound statement, "It will either snow today or it will not snow today." The two statements together will always be true, so before we subject it to a truth table, know that it is a tautology.

*[construct four-row, three-column truth table for the two conditions, first row with title Truth Table for p ∨ ~p, second row begins three columns. Second row p; ~p; p ∨ ~p. Third row T; F; T. Fourth row F; T; T]*

No matter what, the compound statement always leads to a true result, so the statement is a tautology.

Our examples, "I will give you $5 or I will not give you $5," and "It will either snow today or it will not snow today," are very simple.

What about a logic statement that is a bit more complicated? In fact, what if we did not have even the English words, but started with just the symbols?

$(p\wedge q)\to p$

We hope you can "decode" that without the words, but just in case the pure logic evades you, it essentially says this:

*If the truth of Proposition $p$ and Proposition $q$ together is true, then Proposition $p$ is true.*

The truth table for this must have columns for $p$, for $q$, for $(p\wedge q)$, and a fourth column for $(p\wedge q)\to p$

*[construct six-row, four-column truth table with top row titled "Truth Table for (p ∧ q) →p; the second row for identifiers for p, for q, for (p ∧ q), and (p ∧ q) →p; then the four permutations (row three) T T T T; (row four) T F F T; (row five) F T F T; (row six) F F F T]*

No matter what we find with Propositions $p$, $q$, and $(p\wedge q)$, we end up with truth, so this is a tautology. If even one of the final column's findings was false, then we would not have a tautology.

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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