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2.2 Logic

2.2 Logic. Objectives. Determine truth values of conjunctions and disjunctions Construct truth tables Construct and interpret Venn Diagrams. Truth Values. A statement is any sentence that is either true or false, but not both. The truth or falsity of a statement is its truth value .

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2.2 Logic

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  1. 2.2 Logic

  2. Objectives • Determine truth values of conjunctions and disjunctions • Construct truth tables • Construct and interpret Venn Diagrams

  3. Truth Values • A statement is any sentence that is either true or false, but not both. The truth or falsity of a statement is its truth value. • Statements are most often represented using a letter such as p or q. Example:p: Denver is the capital of Colorado.

  4. Truth Values • The negation of a statement has the opposite meaning as well as the opposite truth value of the original statement. Example (using the previous statement): not p also written as ~ p: Denver is not the capital of Colorado.

  5. Truth Values • A compound statement is the joining of two or more statements. Example: p: Denver is a city in Colorado.q: Denver is the capital of Colorado.p and q: Denver is a city in Colorado, and Denver is the capital of Colorado.

  6. Truth Values • When we join two statements with the word “and” as in the previous example we have created a conjunction. • We write conjunctions as p ^ q, which is read as “p and q.” • A conjunction is true only when BOTH statements in it are true.

  7. Example 1a: Use the following statements to write a compound statement for the conjunction p and q. Then find its truth value.p: One foot is 14 inches.q: September has 30 days.r: A plane is defined by three noncollinear points. Answer: One foot is 14 inches, and September has 30 days. p and q is false, because p is false and q is true.

  8. Use the following statements to write a compound statement for the conjunction . Then find its truth value.p: One foot is 14 inches.q: September has 30 days.r: A plane is defined by three noncollinear points. Answer: A plane is defined by three noncollinear points, and one foot is 14 inches. is false, because r is true and p is false. Example 1b:

  9. Use the following statements to write a compound statement for the conjunction . Then find its truth value.p: One foot is 14 inches.q: September has 30 days.r: A plane is defined by three noncollinear points. Answer: September does not have 30 days, and a plane is defined by three noncollinear points. is false because is false and r is true. Example 1c:

  10. Use the following statements to write a compound statement for the conjunction p r. Then find its truth value.p: One foot is 14 inches.q: September has 30 days.r: A plane is defined by three noncollinear points. Example 1d: Answer: A foot is not 14 inches, and a plane is defined by three noncollinear points. ~p r is true, because ~p is true and r is true.

  11. Use the following statements to write a compound statement for each conjunction. Then find its truth value.p: June is the sixth month of the year.q: A square has five sides.r: A turtle is a bird.a. p and r b. Your Turn: Answer: June is the sixth month of the year, and a turtle is a bird; false. Answer: A square does not have five sides, and a turtle is not a bird; true.

  12. Use the following statements to write a compound statement for each conjunction. Then find its truth value.p: June is the sixth month of the year.q: A square has five sides.r: A turtle is a bird. c. d. Your Turn: Answer: A square does not have five sides, and June is the sixth month of the year; true. Answer: A turtle is not a bird, and a square has five sides; false.

  13. More About Truth Values • Statements can also be joined by the word “or.” We call these disjunctions and write them as p Vq, which is read as “p or q.” Example: p: Susan has 1st lunch. q: Susan has 2nd lunch. p Vq : Susan has 1st lunch, or Susan has 2nd lunch.

  14. More About Truth Values • A disjunction is true if at least one of the statements is true. The truth value of a disjunction is only false if both of the statements are false.

  15. Use the following statements to write a compound statement for the disjunction p or q. Then find its truth value. p: is proper notation for “lineAB.”q: Centimeters are metric units.r: 9 is a prime number. Answer: is proper notation for “line AB,” or centimeters are metric units. p or q is true because q is true. It does not matter that p is false. Example 2a:

  16. Use the following statements to write a compound statement for the disjunction . Then find its truth value. p: is proper notation for “lineAB.”q: Centimeters are metric units.r: 9 is a prime number. Answer: Centimeters are metric units, or 9 is a prime number. is true because q is true. It does not matter that r is false. Example 2b:

  17. Use the following statements to write a compound statement for each disjunction. Then find its truth value.p: 6 is an even number.q: A cow has 12 legsr: A triangle has 3 sides.a. p or r b. Your Turn: Answer: 6 is an even number, or a triangle as 3 sides; true. v Answer: A cow does not have 12 legs, or a triangle does not have 3 sides; true.

  18. Venn Diagrams • Often we illustrate conjunctions and disjunctions by using Venn Diagrams. • The Venn Diagram to the right represents the number of students enrolled in each of the electives.

  19. Venn Diagrams In a Venn Diagram the conjunction is represented by the intersection of all sets, (i.e. the white section of 9 students). Meanwhile, a disjunction is simply represented by the union of all the sets, (i.e. all of the circles and intersections).

  20. Example 3: DANCING The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes.

  21. Example 3a: How many students are enrolled in all three classes? The students that are enrolled in all three classes are represented by the intersection of all three sets. Answer: There are 9 students enrolled in all three classes

  22. Example 3b: How many students are enrolled in tap or ballet? The students that are enrolled in tap or ballet are represented by the union of these two sets. Answer: There are 28 + 13 + 9 + 17 + 25 + 29 or 121 students enrolled in tap or ballet.

  23. Example 3c: How many students are enrolled in jazz and ballet and not tap? The students that are enrolled in jazz and ballet and not tap are represented by the intersection of jazz and ballet minus any students enrolled in tap. Answer: There are 25 students enrolled in jazz and ballet and not tap.

  24. Your Turn: PETS The Venn diagram shows the number of students at Mustang Mid-High that have dogs, cats, and birds as household pets.

  25. Your Turn: a. How many students in Mustang Mid-High have at least one of three types of pets? b. How many students have dogs or cats? c. How many students have dogs, cats, and birds as pets? Answer: 311 Answer: 280 Answer: 10

  26. q ~p p ~p Truth Tables • Last, a convenient method for organizing truth values of statements is to use truth tables.

  27. Truth Tables • By constructing truth tables, you can organize the truth values for statement (p), its negation(~ p), any conjunctions of the statement (p ^ q), any disjunctions of the statement (p vq), and even any negations of conjunctions (~ p ^ ~ q ) or any negations of disjunctions (~ p v~q).

  28. Construct a truth table for . Step 1Make columns with the headingsp, q, ~p, and ~p q ~p p ~p Example 4a:

  29. Construct a truth table for . p q ~p ~p T T T F F T F F Example 4a: Step 2List the possible combinations of truth values for p and q.

  30. Construct a truth table for . p q ~p ~p T T F T F F F T T F F T Example 4a: Step 3Use the truth values of p to determine the truth values of ~p.

  31. Construct a truth table for . p q ~p ~p T T F T T F F F F T T T F F T T Example 4a: Step 4Use the truth values for ~p and q to write the truth values for ~p q. Answer:

  32. Construct a truth table for . p q r ~q ~q r p (~q  r) Example 4b: Step 1Make columns with the headingsp, q, r, ~q, ~q r, and p  (~q  r).

  33. Construct a truth table for . p q r ~q ~q r p (~q  r) T T T T F T T T F T F F F T T F F T F T F F F F Example 4b: Step 2List the possible combinations of truth values for p, q, and r.

  34. Construct a truth table for . p q r ~q ~q r p (~q  r) T T T F T F T T T T F F T F F T F T T F F F T T F T F F F F F T Example 4b: Step 3Use the truth values of q to determine the truth values of ~q.

  35. Construct a truth table for . p q r ~q ~q r p (~q  r) T T T F F T F T T T T T F F F T F F T F F T T F F F F T T T F T F F F F F F T F Example 4b: Step 4Use the truth values for ~q and r to write the truth values for ~q r.

  36. Construct a truth table for . 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 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 Example 4b: Step 5Use the truth values for p and ~qr to write the truth values for p (~q  r). Answer:

  37. p q r ~r p q (p q)  ~r Example 4c: Construct a truth table for (p q)  ~r. Step 1Make columns with the headingsp, q, r, ~r, p q, and (p  q)  ~r.

  38. p q r ~r p q (p q)  ~r T T T T F T T T F T F F F T T F F T F T F F F F Example 4c: Construct a truth table for (p q)  ~r. Step 2List the possible combinations of truth values for p, q, and r.

  39. p q r ~r p q (p q)  ~r T T T F T F T F T T F T T F F T F T T F F F T F F T F T F F F T Example 4c: Construct a truth table for (p q)  ~r. Step 3Use the truth values of r to determine the truth values of ~r.

  40. p q r ~r p q (p q)  ~r T T T F T T F T F T T T F T T T F F T T F T T F T F F T F F F T F T T F F F T F Example 4c: Construct a truth table for (p q)  ~r. Step 4Use the truth values for p and q to write the truth values for p q.

  41. p q r ~r p q (p q)  ~r T T T F T F T F T F T F T T F T T T T F F T T T F T T F T F F F T F F F F T F T T T F F F T F F Example 4c: Construct a truth table for (p q)  ~r. Step 5Use the truth values for p q and ~r to write the truth values for (p q)  ~r. Answer:

  42. 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 Your Turn: Construct a truth table for the following compound statement. Answer:

  43. 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 Your Turn: Construct a truth table for the following compound statement. Answer:

  44. 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 Your Turn: Construct a truth table for the following compound statement. Answer:

  45. Assignment • Geometry: Pg. 72 – 73 #4 – 10, 15 - 32, 41 - 44 • Pre-AP Geometry: Pg. 72 – 73 #18 – 32, 34, 36, 38, 40 – 44, 51 and 52

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