CHAPTER 2: DEDUCTIVE REASONING

Section 2-1: If-Then Statements; Converses

IF-THEN STATEMENT

• If-then statements, which are also called conditional statements or conditionals, are statements that include a hypothesis followed by a conclusion.

EXAMPLES OF IF-THEN

1. If it rains after school, then I will give you a ride home.

2. If B is between A and C, then AB + BC = AC.

3. If I don’t pay attention in this class, then I will fail.

4. I will become a 49ers fan if the Raiders move back to L.A.

REPRESENTING AN IF-THENTo represent if-then statements, we let p

represent the hypothesis and let q represent the conclusion.

Using p = hypothesis and q = conclusion, the base form of an if-then statement can be shown by:

If p, then q.

p: hypothesis q: conclusion

CONVERSE

Converse: Reversing something as in position or order.

The converse of a conditional is formed by interchanging the hypothesis and the conclusion.

CONVERSE EXAMPLE

Statement:

If p, then q.

Converse:

If q, then p.

In short, flip the order of the hypothesis and conclusion to find the converse of a statement.

PRACTICE

State the converse of each conditional and tell whether each converse is true or false:

1. If points lie in 1 plane, then they are coplanar.

2. If points are collinear, then they all lie in one line.

3. If x = -2, then 4x = -8.

STATEMENT VS. CONVERSE

• Statements and their converses do not say the same thing.

• Some true statements have converses that are not true.

Example

Statement: If I live in Castro Valley, then I live in the Bay Area.

Converse: If I live in the Bay Area, then I live in Castro Valley. (False)

COUNTEREXAMPLE

• An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false.

• Such an example is called a counterexample.

• Take the converse from the previous slide for example. I could live in the Bay Area but live in another city such as San Leandro.

PRACTICE

Provide a counterexample to show that each statement is false.

1. If ab < 0, then a < 0.

2. If a four-sided figure has four right angles, then it has four congruent sides.

3. If a line lies in a vertical plane, then the line is vertical.

IF-THEN/CONDITIONAL STATEMENTS

• Conditional Statements or If-Then statements are not always written with the words if and then.

General Form Example

If p, then q. If 4x = 12, then x = 3

p implies q. 4x = 12 implies x = 3

p only if q. 4x = 12 only if x = 3

q if p. x = 3 if 4x = 12

BICONDITIONAL

• If a conditional statement and its converse are both true, they can be combined into a single statement using the words “if and only if”.

• Statements that contain the words “if and only if” are called biconditionals.

BICONDITIONAL

The basic form of a biconditional can be found by:

p if and only if q.

PRACTICE

Write the pair of conditionals as a biconditional:

• If B is between A and C, then AB + BC = AC.• If AB + BC = AC, then B is between A and C.

B is between A and C if and only if

AB + BC = AC

CLASSWORK/HOMEWORK

Classwork

•Pg. 34, Classroom Exercises 2-16 even

Homework

•Pg. 35, Written Exercises 2-28 even