Vocabulary Statement--A sentence that is either true or false, but not both. Truth value--The true or false nature of a statement Compound statement--Two.

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Presentation transcript:

Vocabulary Statement--A sentence that is either true or false, but not both. Truth value--The true or false nature of a statement Compound statement--Two or more statements that are connected by the words and or or.

Example 1 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.

A.A B.B C.C D.D Example 1 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. 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 B.B C.C D.D Example 1 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. 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.

Example 2 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 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. 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.

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. ___

A.A B.B C.C D.D Example 2 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. 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.A B.B C.C D.D Example 2 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. 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 B.B C.C D.D Example 2 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. 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.

Concept Truth table--A convenient way to organize truth values.

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

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

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

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

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

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

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

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

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

A.A B.B C.C D.D Example 3 A. TB. TC. TD. T FFFF FTFT FFFF TTFT FFFF TTFF FFFF A. Which sequence of Ts and Fs would correctly complete the last column of the following truth table for the given compound statement? (p  q)  (q  r)

A.A B.B C.C D.D Example 3 A. TB. TC. TD. T TTFT TTTT FTFF TTTT FTFT TTFT FFFF B. Which sequence of Ts and Fs would correctly complete the last column of the following truth table for the given compound statement? (p  q)  (q  r)

Example 4 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 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 or 121 students enrolled in tap or ballet.

Example 4 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 and not tap are represented by the intersection of jazz and ballet minus any students enrolled in tap. Answer:There are – 9 or 25 students enrolled in jazz and ballet and not tap.

A.A B.B C.C D.D Example 4 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 at least one of three types of pets? A.226 B.311 C.301 D.110 Pets

A.A B.B C.C D.D Example 4 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 and/or cats? A.57 B.242 C.252 D.280 Pets

A.A B.B C.C D.D Example 4 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 (all three) as pets? A.10 B.85 C.116 D.311 Pets