1. Lesson 2-2.B Logic: Truth Tables

2. 5-Minute Check on Lesson 2-2.A Transparency 2-2 Make a conjecture about the next item in the sequence. 1. 1, 4, 9, 16, 25 2. 2/3, 3/4, 4/5, 5/6, 6/7 Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 3.Given:  ABC with m  A = 60, m  B = 60 and m  C = 60. Conjecture:  ABC is equilateral. 4. Given:  1 and  2 are supplementary angles. Conjecture:  1 and  2 are congruent. 5. Given:  RST is isosceles. Conjecture: RS  ST 6. Make a conjecture about the next item in the sequence: 64, – 32, 16, – 8, 4. Standardized Test Practice: ACBD –4–4 –2–2 2 4

3. Objectives Determine truth values of conjunctions and disjunctions Construct truth tables

4. Vocabulary And symbol (  ) Or symbol (  ) Not symbol (~) Statement – any sentence that is either true or false, but not both Truth value – the truth or falsity of a statement Negation – has the opposite meaning of the statement, and the opposite truth value Compound statement – two or more statements joined together Conjunction – compound statement formed by joining 2 or more statements with “and” Disjunction – compound statement formed by joining 2 or more statements with “or”

5. Using the following statements: p: One meter is exactly 3 feet. q: December has 31 days. r: Two points define a line. Answer: One meter is exactly 3 feet, and December has 31 days. p and q is false, because p is false and q is true. Write a compound statement for the conjunction p and q, and find its truth value. Write a compound statement for the conjunction r  p, and find its truth value. Answer: Two points define a line, and one meter is exactly 3 feet. r  p is false, because r is true and p is false.

6. Using the following statements: p: One meter is exactly 3 feet. q: December has 31 days. r: Two points define a line. Write a compound statement for the conjunction -q  r, and find its truth value. Answer: December does not have 31 days, and two points define a line. ~q  r is false, because -q is false and r is true. Write a compound statement for the conjunction -p  r, and find its truth value. Answer: One meter is not exactly 3 feet, and two points define a line. ~p  r is true, because -p is true and r is true.

7. Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: January is the first month of the year. q: An octagon has eighty sides. r: A chimpanzee is a dinosaur. Answer: An octagon does not have eighty sides, and January is the first month of the year; true. Answer: A chimpanzee is not a dinosaur, and an octagon has eighty sides; false. c. ~q  p d. ~r  q

8. T T F F F T F T F F T T ~p  q ~p~p qp Step 4 Use the truth values for ~p and q to write the truth values for ~p  q. T T F T Answer: Construct a truth table for ~p  q. ((not p) or q) Step 3 Use the truth values of p to determine the truth values of ~p. Step 2 List the possible combinations of truth values for p and q. Step 1 Make columns with the headings p, q, ~p, and ~p  q.

9. FFTFFF F T F T T T T p  (~q  r) F T F F F T F ~q  r~q  r F T F T F T F ~q F T T F F T T r TF TT FT TF FF FT TT qp Step 5 Use the truth values for p and ~q  r to write the truth values for p  (~q  r). Answer: Construct a truth table for p  (~q  r). (p or ( not q and r )) Step 4 Use the truth values for ~q and r to write the truth values for ~q  r. Step 1 Make columns with the headings p, q, r, ~q, ~q  r, and p  (~q  r). Step 2 List the possible combinations of truth values for p, q, and r. Step 3 Use the truth values of q to determine the truth values of ~q.

10. Step 5 Use the truth values for p  q and ~r to write the truth values for (p  q)  ~r. Answer: Construct a truth table for (p  q)  ~r. ((p or q) and not r) FFTFFF F T F T F T F (p  q)  ~r F T T T T T T p  qp  q F T F T F T F ~r T F T F T F T r FF FT FT TF TF TT TT qp Step 4 Use the truth values for p and q to write the truth values for p  q. Step 3 Use the truth values of r to determine the truth values of ~r. Step 2 List the possible combinations of truth values for p, q, and r. Step 1 Make columns with the headings p, q, r, ~r, p  q, and (p  q)  ~r.

11. Construct a truth table for the following compound statement. a. FFFFFF F F T F T F T F F T F F F T F F F F T F T TFF F T F F T T r TT FT TF TF FT TT qp Answer:

12. b. Answer: FFFFFF T F T F T T T T T T F T T T T F T T T T T TFF F T F F T T r TT FT TF TF FT TT qp Construct a truth table for the following compound statement.

13. c. Answer: FFFFFF T F T T T T T F F T F F F T T F T T T T T TFF F T F F T T r TT FT TF TF FT TT qp Construct a truth table for the following compound statement.

14. Summary & Homework Summary: –Negation of a statement has the opposite truth value of the original statement –Venn diagrams and truth tables can be used to determine the truth values of statements Homework: Day 1: pg 72: 4-17 Day 2: pg 72-3: 18, 19, 25, 26, 35-38, 41-44