Chapter 1 LogicSection 1-1 Statements

• Open your book to page 1 and read the section titled “To the Student”• Now turn to page 3 where we will read the last paragraph• We will use logic to explain mathematical processes you have already used, and some you haven’t• In this book, when a sentence is being discussed for its logical properties, it is written in italics

• The quantifier “for all” is represented by• Universal statements are powerful because they

assert that a certain property holds for every element in a set

• The symbol for “there exists” is • These two symbols will be used frequently• When proving universal statements, you must

prove for all members of the set. You cannot just give an instance (or even a few instances) that works

• To prove an existential statement, you only have to find one instance that works

• Ex1. Is the following statement true or false? Justify your answer. For all real numbers x, x² > x

• Ex2. Is this statement true or false? Justify your answer. a real number x such that x² > x

• Open your book to page 8 and look at the green box at the top

• Stay on this page and we are going to read the rest of the page about universal and existential statements

Section 1-2 Negations• Every statement has a negation which is also a

statement• The symbol for “not p” is • When p is true, the not p is false and vice versa• The most commonly use “not” statement is “It is

not the case that…”• Open your book to page 13 for Example 1• Example 1 shows us that you can take a formal or

an informal approach to negation (unless the directions tell you which approach to take)

p

• The negation of a “for all” statement is a “there exists” statement and vice versa

• The negation of a statement can be generated by reading the statement from left to right and changing to , changing to , and changing p(x, y) to not p(x, y)

• Also, the words “such that” are deleted when is changed to and are added when is changed to

• Ex1. Write the negation of the statement Some rectangles are squares

• Ex2. Negate the statement Everybody loves something

Section 1-3 And and Or and DeMorgan’s Laws

• Real life “if” and “if-not” statements are found in income tax returns

• Ex1. Express x > 3 by writing out each implied and, or, and not

• See the truth tables on page 20• Even though or could be inclusive or exclusive in

the real world, it is always inclusive in math• Logical expressions contain and, or, not, or if-then• The symbol for logical equivalence is ≡

• For the exclusive or people say either a or b but not both (not just either a or b)

• Ex2. The therapeutic range for the dosage L of a high blood pressure medication is given by 1.45 < L < 2.65. Describe the range for which the medication dosage is not therapeutic.

Section 1-5 If-Then Statements• If-then statements are the basis for deduction and

proofs• p implies q is denoted • In the above conditional statement above, p is the

hypothesis (or antecedent) and q is the conclusion (or consequent)

• Open your book to page 32 where we will look at the conditional statement at the bottom of the page

p q

• The only combination of truth values that a true conditional cannot have is a true antecedent and a false consequent (see purple box on page 33)

• The negation of a conditional should be true exactly when the conditional is false

• The negation of a conditional statement is an and-statement

• Open your book to page 34 (Example 2)• Contrapositive: the inference drawn from a

proposition by negating its terms and changing their order, as by inferring “Not B implies not A” from “A implies B.”

• Ex1. Write the negation of the conditional statement: If Tom is a Bengals fan, then he loves the Steelers.

• A conditional and its contrapositive have the same truth values

• Read the blue definition box on page 36• Neither the converse nor the inverse of a

conditional need to have the same truth value as the original conditional

• Ex2. Give the converse and inverse of the universal conditional functions f and real numbers x, if f(2x) = 2f(x), then f(4x) = 4f(x)

Section 1-6 Valid Arguments• The symbol for therefore is • A valid argument is not the same thing as a true

conclusion (the argument could make sense and be “valid” but still be wrong)

• An argument is valid if and only if all its premises are true, then its conclusion is true

• Modus tollens = “method of denial” in Latin• Open your book to page 41 and read the Lewis

Carroll sentences• What do you think?

• Now turn to page 44 and 45 to see how to attack this symbolically

• Ex1. Write the form of the following argument: angles x, if x is a right angle, then the degree measure of x is 90. A particular angle c has degree measure 88. c is not a right angle

• Ex2. Assume that premises (1) and (2) are both true: (1) If my puppy Rover scratches, then Rover has fleas. (2) Rover does not have fleas.What conclusion can be deduced?

Section 1-7 Proofs about Integers• To complete some of the proofs in this lesson, you

will need to use properties from previous classes (Associative, Commutative, Distributive, etc.)

• Proofs in boldface type in this section are ones that the book expects you to be able to write on your own (or ones like it)

• In a direct proof, you go directly from the hypothesis to the conclusion (which is how most proofs are in math)

• Open to page 51 to read the green box

• An even number n is defined by n = 2k• An odd number n is defined by n = 2k + 1• Ex1. Assume that x and y are integers. Show that:

a) is evenb) 12x + 8y + 17 is odd

• Ex2. Prove the following theorem: If m is odd, then m² + 1 is even

5 218x y

Section 1-8 Invalid Arguments• An argument form is invalid if and only if there are

arguments of that form in which the premises are true and the conclusion is false

• A converse error results from confusing the premise with its converse

• An inverse error is an invalid argument of either of the following forms:

p q

Simple formIf p, then q.not p not q

Universal formFor all x, if p(x) then q(x)not p(c), for a particular c not q(c)

• Ex1. Write the form of the argument and determine whether it is valid or invalida) If a number is divisible by 8, then it is divisible by 4. 20 is not divisible by 8. 20 is not divisible by 4b) If it’s Sunday, then Teri reads the comics. Teri reads the comics. It is Sunday.

• If the conclusion is false but the premise is true, this is called an improper induction error

• An invalid argument may have a true conclusion• Only in a valid argument with true premises is the

conclusion guaranteed to be true

Section 1-9 Different Kinds of Reasoning• This section will not be on the test (the last

section in every chapter of this book is like this)• Deductive reasoning uses valid argument forms to

arrive at conclusion and follows strict standards of validity

• Inductive reasoning is a generalization based on the evidence of examples

• When scientists gather data from experiments, they use inductive reasoning (based on what they see in front of them)

• Other types of reasoning to read about in this section: diagnostic reasoning and probabilistic reasoning