Section 3.2: Truth Tables for Negation, Conjunction, and Disjunction

1. Section 3.2: Truth Tables for Negation, Conjunction, and Disjunction Math 121

2. Truth Tables A truth table is used to determine when a compound statement is true or false. They are used to break a complicated compound statement into simple, easier to understand parts.

3. Truth Table for Negation As you can see “P” is a true statement then its negation “~P” or “not P” is false. If “P” is false, then “~P” is true. P ~P T F F T

4. Four Possible Cases When a compound statement involves two simple statements P and Q, there are four possible cases for the combined truth values of P and Q. T T T F T F F F

5. When is a Conjunction True? Suppose I tell the class, “You can retake the last exam and you can turn in this lab late.” Let P be “You can retake the last exam” and Q be “You can turn in this lab late.” Which truth values for P and Q make it so that I kept my promise, P Λ Q to the class?

6. When is a Conjunction True? cont’d. P: “You can retake the last exam.” Q: “You can turn this lab in late.” There are four possibilities. 1. P true and Q true, then P Λ Q is true. 2. P true and Q false, then P Λ Q is false. 3. P false and Q true, then P Λ Q is false. 4. P false and Q false, then P Λ Q is false.

7. Truth Table for Conjunction P ΛQ T T T F T F T F F F F F

8. 3.2 Question 1 What is the truth value of the statement, “U of M is in Ann Arbor and Ann Arbor is in West Virginia”? 1. True 2. False

9. When is Disjunction True? Suppose I tell the class that for this unit you will receive full credit if “You do the homework quiz or you do the lab.” Let P be the statement “You do the homework quiz,” and let Q be the statement “You do the lab.” For which truth values of P and Q would I say that you did what I said, which is PVQ to receive full credit for this unit?

10. When is Disjunction True? cont’d. P: “You do the homework quiz.” Q: “You do the lab.” There are four possibilities: 1. P true and Q true, then P V Q is true. 2. P true and Q false, then P V Q is true. 3. P false and Q true, then P V Q is true. 4. P false and Q false, then P V Q is false.

11. Truth Table for Disjunction P V Q T T T T T F T F T F F F

12. 3.2 Question 2 What is the truth value of the statement, “WVU is in Arizona or Morgantown is in West Virginia”? 1. True 2. False

13. Truth Table Summary You can remember the truth tables for ~(not), Λ(and), and, V(or) by remembering the following: ~(not) - Truth value is always the opposite Λ(and)-Always false, except when both are true V(or) - Always true, except when both are false

14. Making a Truth Table Example Let’s look at making truth tables for a statement involving only ONE ΛorVof simple statements P and Q and possibly negated simple statements ~P and ~Q. For example, let’s make a truth table for the statement ~PVQ

15. Truth Table for ~PVQ Q V P Q ~P T T F T T T F F F F F T T T T F F T F T Final Answer column Opposite of Column 1 Same as Column 2

16. Another Example: P Λ~Q Λ ~Q P Q P T T T F F T F T T T F T F F F F F F T F Final Answer column Same as Column 1 Opposite of Column 2

17. 3.2 Question 3 What is the answer column in the truth table of the statement ~P Λ ~Q ? 1. T 2. T 3. F F F F F T F F F T

18. ~P Λ~Q Λ ~Q P Q ~P T T F F F T F F F T F T T F F F F T T T Final Answer column Opposite ofColumn 1 Opposite of Column 2

19. More Complicated Truth Tables Now suppose we want to make a truth table for a more complicated statement, (PV~Q) V (~PΛQ) We set the truth table up as before. Our final answer will go under the most dominant connective not in parentheses (the one in the middle)

20. More Complicated Truth Tables T T T F F T F T T T F F F T T F T T T F F T T T F F T F Final Answer Opposite ofColumn 1 Opposite ofColumn 2 Same asColumn 1 AND Same asColumn 2 OR

21. More Complicated Truth Tables Now let’s make a truth table for (P V~Q) Λ(~P Λ Q) Each of the statements in parentheses ( P V~Q) and (~P Λ Q) are just like the statements we did previously, so we fill in their truth tables as we just did.

22. More Complicated Truth Tables T T F F F T F T T F F F F T F F T T T F F F T T F F T F Final Answer Opposite ofColumn 1 Opposite ofColumn 2 Same asColumn 1 AND Same asColumn 2 OR

23. Constructing Truth Tables with Three Simple Statements So far all the compound statements we have considered have contained only two simple statements (P and Q), with only four true-false possibilities.

24. Constructing Truth Tables with Three Simple Statements cont’d. When a compound statement consists of three simple statements (P, Q, and R), there are now eight possible true-false combinations.

25. Constructing Truth Tables with Three Simple Statements cont’d.

26. A Three Statement Example Let,s construct a truth table for the statement (P V Q) Λ ~R using the same techniques as before.

27. A Three Statement Example TTTTFFFF TTTTTTFF TTFFTTFF FTFTFTFF FTFTFTFT

28. Practice • Determine the Truth Value for the statement IF: • P is true, Q is false, and R is true (~ P V ~ Q) Λ (~R V ~ P)

29. Practice • Translate into symbols. Then construct a truth table and indicate under what conditions the compound statement is TRUE. • Nathan owns a convertible and Joe does not own a Volvo.

30. Practice • Construct a Truth Table for the following compound statement: R V(P Λ ~ Q)