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Truth Tables and Logic Conjunctions Explained

Learn to determine truth values of conjunctions and disjunctions, construct truth tables, and decipher compound statements using logic symbols. Practice with different scenarios and statements.

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Truth Tables and Logic Conjunctions Explained

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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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