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Lesson 2-2.B

Lesson 2-2.B. Logic: Truth Tables. Transparency 2-2. 5-Minute Check on Lesson 2-2.A. 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

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Lesson 2-2.B

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

  2. Transparency 2-2 5-Minute Check on Lesson 2-2.A • 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. • 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: A B C D 4 –4 2 –2

  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. Write a compound statement for the conjunction p and q, and find its truth value. 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 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. c. ~qp d. ~rq 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.

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

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

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

  11. a. p q r T T T T T T T F T F F F T T F T F T T F F F F F F T T F T T F F T F F F F T F F F F F F F F F F Construct a truth table for the following compound statement. Answer:

  12. b. p q r T T T T T T T F T T T T T T F T T T T F F T F F F T T T T T F F T F T F F T F T T T F F F F F F Construct a truth table for the following compound statement. Answer:

  13. c. p q r T T T T T T T F T T F T T T F T F T T F F T F T F T T T T T F F T F F F F T F T F T F F F F F F Construct a truth table for the following compound statement. Answer:

  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

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