Ch.2 Reasoning and Proof

Pages 60 - 123

2-1 Inductive Reasoning and Conjecture (p.62)

- A conjecture is an educated guess based on known information.

- Inductive reasoning uses a number of specific examples to arrive at a plausible generalization or prediction.

- Counterexample - an example used to show that a given statement is not always true.

Example:

Find the pattern.

2 , 4 , 12 , 48 , 240

The numbers increase in the following order: multiply by 2, multiply by 3, multiply by 4, and multiply by 5.

Conjecture:

The next number will increase when you multiply by 6.

So, it will be 240 • 6 or 1440.

Example:

Find the pattern.

1 , 3 , 6 , 10 , 15 ,

The numbers increase by 2, 3, 4, and 5.

Conjecture:

The next number will increase by 6.

So, it will be 15 + 6 or 21.

Example:

Is the conjecture true or false? If false, give a counterexample.

1. Given: S, T, and U are collinear and ST = TU.

Conjecture: T is the midpoint of segment SU.

True

2. Given: Angle 1 and Angle 2 are adjacent angles.

Conjecture: Angle 1 and Angle 2 form a linear pair.

False; Angle 1 and Angle 2 could each measure 60 degrees.

(Sample answer)

PQ = SR, QR = PS

- PQRS is a rectangle.

Make a conjecture based on the given information.

Draw a figure to illustrate your conjecture, if necessary.

Example:

2-2 Logic (p.67)

- Statement - any sentence that is either true or false, but not both.

- Truth value - The truth or falsity of a statement.

- Negation - If a statement is represented by p,

then not p is the negation of the statement.

~p is read as not p

* The negation of a statement has the opposite meaning as well as an opposite truth value.

p: Halloween is on October 31.

(This statement is true.)

not p (~p): Halloween is not on October 31.

(This statement is false.)

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

Example:

- Compound statement - a statement formed by joining two or more statements.

Consider:

p: B-18 is a room in the high school. (true)

q: B-18 is Mrs. Bannon’s room. (true)

p and q: B-18 is a room in the high school, and

B-18 is Mrs. Bannon’s room.

The statement formed using and is an example of a conjunction.

- conjunction - a compound statement formed by joining two or more statements with the word and.

p ^ q is read p and q

* A conjunction is true only when both statements in it are true.

- disjunction - a compound statement formed by joining two or more statements with the word or.

p q is read p or q

^

* A disjunction is true if at least one of the statements in it is true.

* ( similar to Example 1, page 68 )

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

p: One foot is 14 inches. (false)

q: September has 30 days. (true)

r: A plane is defined by three noncollinear points. (true)

a. p and q

b. r ^ p

One foot is 14 inches, and September has 30 days.

False

A plane is defined by three noncollinear points, and one foot is 14 inches.

False

c. ~p ^ r

d. ~q ^ r

A foot is not 14 inches, and a plane is defined by three noncollinear points.

True

September does not have 30 days, and a plane is defined by three noncollinear points.

False

Use the following statements to write a compound statement for each disjunction. Then find its truth value. * ( similar to Example 2, page 69 )

p: Vertical angles are formed by perpendicular lines. (true)

q: Centimeters are metric units. (true)

r: 9 is a prime number. (false)

a. p or q

b. q r

^

Vertical angles are formed by perpendicular lines, or centimeters are metric units.

True

Centimeters are metric units, or 9 is a prime number.

True

Venn Diagrams:

Conjunctions and disjunctions can be illustrated with Venn diagrams.

Ex. The Venn diagram shows the number of students enrolled in a Dance School.

Tap

28

Jazz

43

29

Ballet

13

17 259

a. How many students are enrolled in all three classes?

b. How many students are enrolled in tap or ballet?

c. How many students are enrolled in jazz and ballet and not tap?

9

121

25

Truth Table - a table used as a convenient method for organizing the truth values of statements.

*A conjunction is true only when both statements are true.

*A disjunction is false only when both statements are false.

r q r ^ q

T T T

T F F

F T F

F F F

Conjunction Disjunction

r q r q

T T T

T F T

F T T

F F F

r ~r

T F

F T

Negation^

2-3 Conditional Statements (p.75)

- conditional statement - a statement that can be written in if - then form.

- if - then statement - a compound statement of the form “if A, then B”, where A and B are statements.

- hypothesis - in a conditional statement, the statement that immediately follows the word if.

- conclusion - in a conditional statement, the statement that immediately follows the word then.

Ex.] Get $1500 Cash Back When You Buy a New Car

Ex.] If you buy a car, then you get $1500 cash back.

p q , read if p then q , or p implies q.

Converse, Inverse, and Contrapositive (p.77)

(chart)

- related conditionals - statements such as the converse, inverse, and contrapositive that are based on a given conditional statement.

- converse - the statement formed by exchanging the hypothesis and conclusion of a conditional statement

- inverse - the statement formed by negating both the hypothesis and conclusion of a conditional statement

- contrapositive - the statement formed by negating both the hypothesis and conclusion of the converse a conditional statement

- logically equivalent -statements that have the same truth values.

p q Conditional

p q

Converse

q p

Inverse

~p ~q

Contrapositive

~q ~p

T T T T T T

T F F T T F

F T T F F T

F F T T T T

(p.77)

Example:Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If false, give a counterexample.

Conditional:

Converse:

Inverse:

Contrapositive:

If a shape is a square, then it is a rectangle.

If a shape is a rectangle, then it is a square.

False ; a rectangle with a length of 2 and a width of 4 is not a square.

If a shape is not a square, then it is not a rectangle.

False ; a 4-sided polygon with side length 2, 2, 4, and 4 is not a square.

If a shape is not a rectangle, then it is not a square.

True

- biconditional - the conjunction of a conditional statement and its converse.

(p q) (q p) is written (p q) and read p if and only if q

* If and only if can be abbreviated iff.

(p. 81)

2-4 Deductive Reasoning (p.82)

- deductive reasoning - a system of reasoning that uses facts, rules, definitions, or properties to reach logical conclusions

- Law of Detachment - if p q is a true conditional and p is true, then q is also true.

[(p q) p] q

Ex.] A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make?

The garden will be watered.

* Tip: Label the hypotheses and conclusions of a series of statements before applying the Law of Syllogism.

(examples on pages 82 - 83.)

Ex.] Use the Law of Syllogism to draw a conclusion from the following statements: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. If a quadrilateral is a square, then it is a rectangle.

- Law of Syllogism - if p q and q r are true conditionals, then p r is also true. (similar to the Transitive Property of Equality)

[(p q) (q r )] (p r)

Ex.] For the given statement, what can you conclude? Given: If A is acute, mA < 90.

A is acute. Conclusion: mA < 90

Valid ; Law of Detachment

Ex.] Does the following argument illustrate the Law of Detachment? Given: If a road is icy, then driving conditions are hazardous.

Driving conditions are hazardous.

Invalid: you do not know that the hypothesis is true.

Determine whether the statements are valid and if so, which Law is being used.