Splash Screen

Five-Minute Check (over Lesson 2–1) Then/Now New Vocabulary Example 1: Truth Values of Conjunctions Example 2: Truth Values of Disjunctions Concept Summary: Negation, Conjunction, Disjunction Example 3: Construct Truth Tables Example 4: Real-World Example: Use Venn Diagrams Lesson Menu

Make a conjecture about the next item in the sequence. 1, 4, 9, 16, 25 D. 40 5-Minute Check 1

Make a conjecture about the next item in the sequence. D. 5-Minute Check 2

Determine whether the conjecture is true or false Determine whether the conjecture is true or false. Given: ΔABC, if mA = 60, mB = 60, and mC = 60. Conjecture: ΔABC is an equilateral triangle. A. true B. false 5-Minute Check 3

Determine whether the conjecture is true or false Determine whether the conjecture is true or false. Given: 1 and 2 are supplementary angles. Conjecture: 1 and 2 are congruent. A. true B. false; m1 = 70 and m2 = 110 5-Minute Check 4

Determine whether the conjecture is true or false Determine whether the conjecture is true or false. Given: ΔRST has two congruent sides. Conjecture: A. true B. false; 5-Minute Check 5

Find the next two terms in the sequence 243, –81, 27, –9, .... A. –3, –1 B. 3, –1 C. 3, 1 D. –3, 1 5-Minute Check 6

You found counterexamples for false conjectures. Determine truth values of negations, conjunctions, and disjunctions, and represent them using Venn diagrams. Find counterexamples. Then/Now

statement truth value negation compound statement conjunction disjunction truth table Vocabulary

Truth Values of Conjunctions A. 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: p and q: One foot is 14 inches, and September has 30 days. Although q is true, p is false. So, the conjunction of p and q is false. Example 1

Truth Values of Conjunctions B. 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. Answer: ~p  r: 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. Example 1

A. A square has five sides and a turtle is a bird; false. A. Use the following statements to write a compound statement for p and r. 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. A square has five sides and a turtle is a bird; false. B. June is the sixth month of the year and a turtle is a bird; true. C. June is the sixth month of the year and a square has five sides; false. D. June is the sixth month of the year and a turtle is a bird; false. Example 1

A. A square has five sides and a turtle is not a bird; true. B. Use the following statements to write a compound statement for ~q  ~r. 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. A square has five sides and a turtle is not a bird; true. B. A square does not have five sides and a turtle is not a bird; true. C. A square does not have five sides and a turtle is a bird; false. D. A turtle is not a bird and June is the sixth month of the year; true. Example 1

Truth Values of Disjunctions A. 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 “segment AB.” q: Centimeters are metric units. r: 9 is a prime number. Answer: is proper notation for “segment AB,” or centimeters are metric units. Both p and q are true, so p or q is true. Example 2

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

Truth Values of Disjunctions C. Use the following statements to write a compound statement for the disjunction ~p  r. Then find its truth value. p: is proper notation for “segment AB.” q: Centimeters are metric units. r: 9 is a prime number. Answer: AB is not proper notation for “segment AB,” or 9 is a prime number. Since not p and r are both false, ~p  r is false. ___ Example 2

A. 6 is an even number or a cow has 12 legs; true. A. Use the following statements to write a compound statement for p or r. Then find its truth value. p: 6 is an even number. q: A cow has 12 legs r: A triangle has 3 sides. A. 6 is an even number or a cow has 12 legs; true. B. 6 is an even number or a triangle has 3 sides; true. C. A cow does not have 12 legs or 6 is an even number; true. D. 6 is an even number or a triangle does not have 3 side; true. Example 2

B. A cow has 12 legs or a triangle has 3 sides; true. B. Use the following statements to write a compound statement for ~q  ~r. Then find its truth value. p: 6 is an even number. q: A cow has 12 legs. r: A triangle has 3 sides. A. A cow does not have 12 legs or a triangle does not have 3 sides; true. B. A cow has 12 legs or a triangle has 3 sides; true. C. 6 is an even number or a triangle has 3 sides; true. D. A cow does not have 12 legs and a triangle does not have 3 sides; false. Example 2

C. Use the following statements to write a compound statement for ~p  q. Then find its truth value. p: 6 is an even number. q: A cow has 12 legs. r: A triangle has 3 sides. A. 6 is an even number or a cow has 12 legs; true. B. 6 is not an even number or a cow does not have 12 legs; true. C. A cow does not have 12 legs, or a triangle has 3 sides; true. D. 6 is not an even number or a cow has 12 legs; false. Example 2

Concept

A. Construct a truth table for ~p  q. Construct Truth Tables A. Construct a truth table for ~p  q. Step 1 Make columns with the heading p, q, ~p, and ~p  q. Example 3

A. Construct a truth table for ~p  q. Construct Truth Tables A. Construct a truth table for ~p  q. Step 2 List the possible combinations of truth values for p and q. Example 3

A. Construct a truth table for ~p  q. Construct Truth Tables A. Construct a truth table for ~p  q. Step 3 Use the truth values of p to determine the truth values of ~p. Example 3

A. Construct a truth table for ~p  q. Construct Truth Tables A. Construct a truth table for ~p  q. Step 4 Use the truth values of ~p and q to write the truth values for ~p  q. Answer: Example 3

B. Construct a truth table for p  (~q  r). Construct Truth Tables B. Construct a truth table for p  (~q  r). Step 1 Make columns with the headings p, q, r, ~q, ~q  r, and p  (~q  r). Example 3

B. Construct a truth table for p  (~q  r). Construct Truth Tables B. Construct a truth table for p  (~q  r). Step 2 List the possible combinations of truth values for p, q, and r. Example 3

B. Construct a truth table for p  (~q  r). Construct Truth Tables B. Construct a truth table for p  (~q  r). Step 3 Use the truth values of q to determine the truth values of ~q. Example 3

B. Construct a truth table for p  (~q  r). Construct Truth Tables B. Construct a truth table for p  (~q  r). Step 4 Use the truth values for ~q and r to write the truth values for ~q  r. Example 3

B. Construct a truth table for p  (~q  r). Construct Truth Tables B. Construct a truth table for p  (~q  r). Step 5 Use the truth values for ~q  r and p to write the truth values for p  (~q  r). Answer: Example 3

A. Which sequence of Ts and Fs would correctly complete the last column of the truth table for the given compound statement? (p  q)  (q  r) A. T B. T C. T D. T F F F F F T F T F F F F T T F T F F F F T T F F F F F F Example 3

B. Which sequence of Ts and Fs would correctly complete the last column of the truth table for the given compound statement? (p  q)  (q  r) A. T B. T C. T D. T T T F T T T T T F T F F T T T T F T F T T T F T F F F F Example 3

A. How many students are enrolled in all three classes? Use Venn Diagrams DANCING The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes. A. 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. Example 4

B. How many students are enrolled in tap or ballet? Use Venn Diagrams DANCING The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes. B. 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. Example 4

C. How many students are enrolled in jazz and ballet, but not tap? Use Venn Diagrams DANCING The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes. C. How many students are enrolled in jazz and ballet, but not tap? The students that are enrolled in jazz and ballet, but not tap, are represented by the intersection of jazz and ballet minus any students enrolled in tap. Answer: There are 25 + 9 – 9 or 25 students enrolled in jazz and ballet, but not tap. Example 4

A. How many students in Manhattan School have a dog, a cat, or a bird? Pets PETS The Venn diagram shows the number of students at Manhattan School that have dogs, cats, and birds as household pets. A. How many students in Manhattan School have a dog, a cat, or a bird? A. 226 B. 311 C. 301 D. 110 Example 4

B. How many students have dogs or cats? Pets PETS The Venn diagram shows the number of students at Manhattan School that have dogs, cats, and birds as household pets. B. How many students have dogs or cats? A. 57 B. 242 C. 252 D. 280 Example 4

C. How many students have dogs, cats, and birds as pets? PETS The Venn diagram shows the number of students at Manhattan School that have dogs, cats, and birds as household pets. C. How many students have dogs, cats, and birds as pets? A. 10 B. 85 C. 116 D. 311 Example 4

End of the Lesson