logic statement-any sentence that is either true or false statement-any sentence that is either true...

LogicLogic

Statement-any sentence that is Statement-any sentence that is either true or falseeither true or false

Truth value-the truth or falsity of a Truth value-the truth or falsity of a statementstatement

Negation-has opposite meaning as Negation-has opposite meaning as well as opposite truth valuewell as opposite truth value

Compound Statement-two or more Compound Statement-two or more statements joinedstatements joined

LogicLogic

Conjunction (^) – compound Conjunction (^) – compound statement formed by combing two or statement formed by combing two or more statements with the word more statements with the word andand

Disjunction (v) - compound Disjunction (v) - compound statement formed by combing two or statement formed by combing two or more statements with the word ormore statements with the word or

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.

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.

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.

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: 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.

Answer: June is the sixth month of the year, and a turtle is a bird; false.

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.

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.

c.

d.

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.

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 “line AB.”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.

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.

Use the following statements to write a compound statement for the disjunction . Then find its truth value.

p: is proper notation for “line AB.”q: Centimeters are metric units.r: 9 is a prime number.

Answer: 6 is an even number, or a triangle as 3 sides; true.

Answer: A cow does not have 12 legs, or a triangle does not have 3 sides; true.

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.

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

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

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.

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 + 9 – 9 or 25 students enrolled in jazz and ballet and not tap.

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 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

Step 1 Make columns with the headingsp, q, ~p, and ~p

Construct a truth table for .

~~p p ~~ppqqpp

Step 2 List the possible combinations of truth values for p and q.

Construct a truth table for .

FFFFTTFFFFTTTTTT

~~p p ~~ppqqpp

Step 3 Use the truth values of p to determine the truth values of ~p.

Construct a truth table for .

TTFFFFTTTTFFFFFFTTFFTTTT

~~p p ~~ppqqpp

Step 4 Use the truth values for ~p and q to write the truth values for ~p q.

Answer:

Construct a truth table for .

TTTTFFFFTTTTTTFFFFFFFFTTTTFFTTTT

~~p p ~~ppqqpp

pp ( (~~qq rr))~~qq rr~q~qrrqqpp

Step 1 Make columns with the headingsp, q, r, ~q, ~q r, and p (~q r).

Construct a truth table for .

Step 2 List the possible combinations of truth values for p, q, and r.

Construct a truth table for .

FFFFFF

pp ( (~~qq rr))~~qq rr~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Step 3 Use the truth values of q to determine the truth values of ~q.

Construct a truth table for .

TTFFFFFF

pp ( (~~qq rr))~~qq rr

FF

TT

FF

TT

FF

TT

FF

~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Step 4 Use the truth values for ~q and r to write the truth values for ~q r.

Construct a truth table for .

FFTTFFFFFF

pp ( (~~qq rr))

FF

TT

FF

FF

FF

TT

FF

~~qq rr

FF

TT

FF

TT

FF

TT

FF

~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Step 5 Use the truth values for p and ~q r to write the truth values for p (~q r).

Answer:

Construct a truth table for .

FFFFTTFFFFFF

FF

TT

FF

TT

TT

TT

TT

pp ( (~~qq rr))

FF

TT

FF

FF

FF

TT

FF

~~qq rr

FF

TT

FF

TT

FF

TT

FF

~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Construct a truth table for (p q) ~r.

((pp qq) ) ~~rrpp qq~r~rrrqqpp

Step 1 Make columns with the headingsp, q, r, ~r, p q, and (p q) ~r.

Step 2 List the possible combinations of truth values for p, q, and r.

FFFFFF

((pp qq) ) ~~rrpp qq~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Construct a truth table for (p q) ~r.

Step 3 Use the truth values of r to determine the truth values of ~r.

Construct a truth table for (p q) ~r.

TTFFFFFF

((pp qq) ) ~~rrpp qq

TT

FF

FF

TT

TT

FF

FF

~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Step 4 Use the truth values for p and q to write the truth values for p q.

Construct a truth table for (p q) ~r.

FFTTFFFFFF

((pp qq) ) ~~rr

TT

FF

TT

TT

TT

TT

TT

pp qq

TT

FF

FF

TT

TT

FF

FF

~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Step 5 Use the truth values for p q and ~r to write the truth values for (p q) ~r.

Answer:

Construct a truth table for (p q) ~r.

FFFFTTFFFFFF

FF

FF

TT

FF

TT

FF

TT

((pp qq) ) ~~rr

TT

FF

TT

TT

TT

TT

TT

pp qq

TT

FF

FF

TT

TT

FF

FF

~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp

Construct a truth table for the following compound statement.a.

FFFFFFFFFFFF

FF

FF

TT

FF

TT

FF

TT

FF

FF

TT

FF

FF

FF

TT

FF

FF

FF

FF

TT

FF

TT

TTFFFF

FF

TT

FF

FF

TT

TT

rr

TTTT

FFTT

TTFF

TTFF

FFTT

TTTT

qqpp

Answer:

b. Answer:

FFFFFFFFFFFF

TT

FF

TT

FF

TT

TT

TT

TT

TT

TT

FF

TT

TT

TT

TT

FF

TT

TT

TT

TT

TT

TTFFFF

FF

TT

FF

FF

TT

TT

rr

TTTT

FFTT

TTFF

TTFF

FFTT

TTTT

qqpp

Construct a truth table for the following compound statement.

c. Answer:

FFFFFFFFFFFF

TT

FF

TT

TT

TT

TT

TT

FF

FF

TT

FF

FF

FF

TT

TT

FF

TT

TT

TT

TT

TT

TTFFFF

FF

TT

FF

FF

TT

TT

rr

TTTT

FFTT

TTFF

TTFF

FFTT

TTTT

qqpp

Construct a truth table for the following compound statement.