# Tautologies Example 1: What do you notice about each sentence below?

 1 A number is even or a number is not even. 2 Cheryl passes math or Cheryl does not pass math. 3 It is raining or it is not raining. 4 A triangle is isosceles or a triangle is not isosceles.

Each sentence in Example 1 is the disjunction of a statement and its negation  Each of these sentences can be written in symbolic form as p ~p. Recall that a disjunction is false if and only if both statements are false; otherwise it is true. By this definition, p ~p is always true, even when statement p is false or statement ~p is false!  This is illustrated in the truth table below:

 p ~p p ~p T F T F T T

The compound statement p ~p consists of the individual statements p and ~p. In the truth table above, p ~p is always true, regardless of the truth value of the individual statements. Therefore, we conclude that p ~p is a tautology.

Definition: A compound statement, that is always true regardless of the truth value of the individual statements, is defined to be a tautology.

Let's look at another example of a tautology.

Example 2: Is (p q) p a tautology?

 p q p q (p q) p T T T T T F F T F T F T F F F T

Solution: The compound statement (p q) p consists of the individual statements p, q, and p q. The truth table above shows that (p q) p is true regardless of the truth value of the individual statements. Therefore, (p q) p is a tautology.

In the examples below, we will determine whether the given statement is a tautology by creating a truth table.

Example 3: Is x (x y) a tautology?

 x y x y x (x y) T T T T T F T T F T T T F F F T

Solution: Yes; the truth values of x (x y) are {T, T, T, T}.

Example 4: Is ~b b a tautology?

 b ~b ~b b T F T F T F

Solution: No; the truth values of ~b b are {T, F}.

Example 5: Is (p q) (p q) a tautology?

 p q (p q) (p q) (p q) (p q) T T T T T T F T F F F T T F F F F F F T

Solution: No; the truth values of (p q) (p q) are {T, F, F, T}.

Example 6: Is [(p q) p] p a tautology?

 p q p q (p q) p [(p q) p] p T T T T T T F F F T F T T F T F F T F T

Solution: Yes; the truth values of [(p q) p] p are {T, T, T, T}.

Example 7: Is (r s) (s r) a tautology?

 r s r s s r (r s) (s r) T T T T T T F F T F F T T F F F F T T T

Solution: No; the truth values of (r s) (s r) are {T, F, F, T}. Summary: A compound statement that is always true, regardless of the truth value of the individual statements, is defined to be a tautology. We can construct a truth table to determine if a compound statement is a tautology.

### Exercises

 1. What is the truth value of r ~r? TrueFalseNot enough information was given.None of the above. RESULTS BOX:
 2. Is the following statement a tautology?  s ~s YesNoNot enough information was given.None of the above. RESULTS BOX:
 3. Is the following statement a tautology? [(p q) ~p] q YesNoNot enough information was given.None of the above. RESULTS BOX:
 4. Is the following statement a tautology? ~(x y) (~x ~y) YesNoNot enough information was given.None of the above. RESULTS BOX:
 5. Is the following statement a tautology? a ~a YesNoNot enough information was givenNone of the above RESULTS BOX: