2. Truth Tables Presented by: Tutorial Services The Math Center

3. Truth Tables A truth table is a device used to determine when a compound statement is true or false. A truth table is a device used to determine when a compound statement is true or false. Formal Name SymbolRead Symbolic Form Negation~“Not”~p Conjunction “ And” p q Disjunction “ Or” p q Conditional“If-then” Bi-conditional “If and only If” p q Connectives used in truth tables:

4. Types of Arguments pq TTT TFF FTF FFF Conjunction When finding the truth value of a conjunction, all values must be true in order for the entire conjunction to be true. For example, if p and q are true, then (p  q) is true. For example, if p is true and q is false, (p  q) is false. For example, if p and q are false, then (p  q) is false.

5. Types of Arguments pq TTT TFT FTT FFF Disjunction When finding the truth value of a disjunction, only one value needs to be true in order for the entire disjunction to be true. For example, if p is true and q is false, then (p  q) is true. For example, if both p and q are true, then (p  q) is true. For example, if both p and q are false, then (p  q) is false.

6. Types of Arguments p~p TF FT Negation The truth values of ~p are exactly the opposite truth values of p. For example, true for p would be false for ~p. For example, false for (p  q) would be true for ~(p  q).

7. Types of Arguments pq p q TTT TFF FTT FFT Conditional When finding the truth value of a conditional statement, same values will be true. Otherwise, follow the truth value of the conclusion (which is the second proposition). For example, if p and q are false, then (p  q) is true. For example, if p is true and q is false, then (p  q) is false. For example, if p is false and q is true, then (p  q) is true.

8. Types of Arguments pq TTT TFF FTF FFT Bi-conditional When finding the truth value of a bi-conditional statement, same values will be true. Otherwise, the truth value will be false. For example, if both p and q are false, then (p  q) is true. For example, if p is true and q is false, then (p  q) is false. For example, if p is false and q is true, then (p  q) is false.

9. Examples pq p  q ~q TTTFT TFFTT FTFFF FFFTF Example1:

10. Examples pqr~r q ~r TTTFFF TTFTTT FFTFFT FTTFFT TFFTFF TFTFFF FTFTTT FFFTFT Example 2:

11. Examples pqr~r q ~r TTTFFF TTFTTT FFTFFT FTTFFT TFFTFF TFTFFF FTFTTF FFFTFT Example 3:

12. Truth Tables Links  Truth Tables Handout Truth Tables Handout Truth Tables Handout