2-2 Logic

You found counterexamples for false conjectures.

• Determine truth values of negations, conjunctions, and disjunctions, and represent them using Venn diagrams.

• Find counterexamples.

Determining Truth Values• A statement is a sentence that is either true or false.

• The truth value of a statement is either true (T) or false (F).

• Statements are often represented using a letter such as p or q.

p: A rectangle is a quadrilateral. Truth value: T

• The negation of a statement has the opposite meaning and the opposite truth value. The negation of the statement above is not p or ~p.

~p: A rectangle is not a quadrilateral. Truth value: F

Conjunction• Compound statements are two or more statements joined

by the word and or or.• A conjunction is a compound statement using the word

and.• A conjunction is true only when both statements that form

it are true.

p: A rectangle is a quadrilateral. Truth value: T

q: A rectangle is convex. Truth value: T

p and q: A rectangle is a quadrilateral, and a rectangle is convex.

• Since both p and q are true, the conjunction p and q, also written p^q, is true.

Truth Values of ConjunctionsA. 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.

Truth Values of ConjunctionsB. 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 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 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.

Disjunction• A disjunction is a compound statement that used the

word or.

p: John studies geometry.

q: John studies chemistry.

p or q: John studies geometry or John studies chemistry.

A disjunction is true if at least one of the statements is true. If John studies either geometry or chemistry or both subjects, the disjunction p or q, also written p ν q, is true. John studies neither geometry nor chemistry, p or q is false.

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.

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.

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.

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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 legsr: 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.

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. 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.

Construct Truth Tables

A. Construct a truth table for ~p q.

Step 1 Make columns with the heading p, q, ~p, and ~p q.

Construct Truth TablesA. Construct a truth table for ~p q.

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

Construct Truth TablesA. Construct a truth table for ~p q.

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

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:

Use Venn Diagrams

• Conjunctions can be illustrated with Venn Diagrams.

p and q: A rectangle is a quadrilateral, and a rectangle is convex.

The Venn diagram shows that a rectangle (R) is located in the intersection of the set of quadrilaterals and the set of convex polygons. The rectangles must be in the set containing quadrilaterals and in the set of convex polygons.

quadrilaterals convexR

A disjunction can be illustrated with a Venn Diagram. The disjunction is represented by the union of the two sets.

The union includes all polygons that are quadrilaterals, convex, or both.

The disjunction includes these three regions:

p ν ~q quadrilaterals that are not convex

~p ν q convex polygons that are not quadrilaterals

p ۸ q polygons that are both quadrilaterals and convex

p ν~q ~p ν qp ۸ q

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.

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.

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

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

Pets

p. 103, 11-14, 17-22, 31, 33,

2-2 Assignment