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CHAPTER 2. By John Cobb. Lesson 1: Conditional Statements. Conditional Statement. If p , then q. Hypothesis The 'if' clause. Conclusion The 'then' clause. Lesson 1: Conditional Statements. If you give a mouse a cookie, then he's going to want some milk

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CHAPTER 2

By John Cobb

Lesson 1: Conditional Statements

Conditional Statement

Ifp,thenq

Hypothesis

The 'if' clause

Conclusion

The 'then' clause

Lesson 1: Conditional Statements

If you give a mouse a cookie,then he's going to want some milk

An equilateral triangle has sides of equal length.

Although that sentence doesn't have an explicit if or then, it still has ahypothesis and conclusion. It can, however, be easily converted into a conventional if-then statement

If a shape is an equilateral triangle, then it has sides of equal length.

Lesson 1: Conditional Statements

Ways to write if p, then q

• if p, q
• q, if p
• p implies q
• p only if q
Lesson 1: Conditional Statements

Euler diagrams

q

he's going to want some milk

conclusion

p

hypothesis

You give a mouse a cookie

If you give a mouse a cookie,then he's going to want some milk

Lesson 2: Definitions

Converse - The converse of conditional statement is found by interchanging its hypothesis and conclusion. In symbols, the converse of p q is q p

If a figure is a hexagon, then it is a polygon with 8 sides

If a figure is a polygon with 8 sides, then it is a hexagon

Lesson 2: Definitions

Biconditional - A statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.”

p q means p q and q p

If and only if can be abbreviated as "iff"

A figure is a hexagon, iff it is a polygon with 8 sides

Lesson 2: Definitions

For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false.

Definition - A definition is a statement that describes a mathematical object and can be written as a true biconditional

Lesson 2.5

The negation of statement p is “not p,” written as ~p. The negation of a true statement is false, and the negation of a false statement is true

The inverse is the statement formed by negating the hypothesis and conclusion

~p ~q

Lesson 2.5

The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion

It is a the "negated converse" of a conditional statement

Lesson 2.5

Related conditional statements that have the same

truth value are called logically equivalent

statements

Logical Equivalents

Conditional Converse

ContrapositiveInverse

Lesson 3: Direct Proof

Syllogism - A syllogism is an argument of the form

a b

b c

Therefore, a c

c

b

a

Lesson 3: Direct Proof

1. If I set my alarm, I will wake up at 7

2. If I wake up at 7, I'll catch the bus

3. If I catch the bus, I won't be late to homeroom

4. If I'm not late to homeroom, I won't get in trouble

5. If I don't get in trouble, I won't get a detention

Therefore,

If I set my alarm, I won't get a detention

Lesson 3: Direct Proof

a b

b c

Therefore, a c

A syllogism is an example of a direct proof

Theorem - A theorem is a statement that is a proved by reasoning deductively from already accepted statements

Premises

Conclusion

Chapter 2 Lab
• The lab in this chapter was all about syllogisms
• It involved arranging scrambled conditional statements into a logical syllogism
• It also involved logical equivalency
Lesson 4: Indirect Proof

In an indirect proof, an assumption is made at the beginning that leads to a contradiction. The contradiction indicates that the assumption is false and desired conclusion is true.

If 5x=25, then x=4

Proof

Suppose that x≠4.

If x≠4, then 5x≠25

This contradicts the fact that 5x=25

Therefore, what we supposed is false, and x=4

Hint

∴ means therefore

Proof for "If a, then c"

Direct Proof

If a, then b.

If b, then c.

Therefore, if a, then c

Indirect Proof

Suppose not c is true.

If not c, then d

If d, then e

(Continue until a contradiction is found)

Therefore, not c is false, so c is true

Lesson 5: A Deductive System

Postulate - A postulate is a statement that is assumed to be true without proof

Postulate 1

Two points determine a line

Postulate 2

Three noncollinear points determine a plane

Hint

Definitions are true as a conditional statement and its converse; Postulates are not

Lesson 6: Some Famous Theorems of Geometry

The Pythagorean Theorem

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides

Lesson 6: Some Famous Theorems of Geometry

The Triangle Angle Sum Theorem

The sum of the angles of a triangle is 180°

Lesson 6: Some Famous Theorems of Geometry

If the diameter of circle is d, its circumference is πd

If the radius of circle is r, its area is πr

2

Conclusion

*If a biconditional is true, it is a definition

Logical Equivalents