1. 3
Logic
The Study of What’s True or False or Somewhere in Between

2. 3.2
Truth Tables
Learn the truth tables for the five fundamental connectives
Compute truth tables for compound statements
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3. 3.2 Truth Tables Determine when statements are logically equivalent
State and apply DeMorgan’s laws.

4. Truth Tables
We want to know if a pair of similar statements such as and mean the same thing.
We use truth tables to determine when compound statements are true and when they are false, and whether a pair of statements have the same meaning.

5. Truth Tables

6. Truth Tables If p is true, then ~p is false.
If p is false, then ~p is true.
(example on next slide)

7. Truth Tables
Example:

8. Truth Tables
A conjunction is true only when both of its components are true.
(example on next slide)

9. Truth Tables
Example:
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10. Truth Tables
Example:
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11. Truth Tables
Solution:

12. Truth Tables A disjunction is false only when both p and q are false.
(example on next slide)

13. Truth Tables
Example:
(solution on next slide)

14. Truth Tables
Solution:

15. Truth Tables
Because of their similar structures, there are parallels that occur in logic and set theory.
Often if a result is true for set theory, then a similar result also holds for logic (and vice versa).

16. Compound Statements
We use truth tables to find the logical values of complex statements.
Example: Compute a truth table for
(solution on next 3 slides)

17. Compound Statements
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18. Compound Statements
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19. Compound Statements
If p is false and q is true, line 3 tells us that is false.

20. Compound Statements
Definition: If the final column of a truth table contains all T’s, then the statement is always true. Such a statement is called a tautology.
Example: “This contestant will win or will not win.”

21. Compound Statements
How many lines should your truth table have?

22. Logically Equivalent Statements
Logically equivalent statements express the same meaning.
(example on next slide)

23. Logically Equivalent Statements
Example:
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24. Logically Equivalent Statements
Solution:
The two statements are logically equivalent.

25. DeMorgan’s Laws
(example on next slide)

26. DeMorgan’s Laws
Example:

27. DeMorgan’s Laws

28. Truth Tables: Alternative Method