Chapter 4-2 Notes

Definitions

• Conditional Statement (If-Then Statement)

– A statement that has a and a

• Hypothesis

– The part of an if-then statement

– Not always first

• Conclusion

– The part of an if-then statement

– Not always second

hypothesis conclusion

first

appearing

second

appearing

Definitions

• Statement 1

–

• Statement 2

–

• Statement 3

–

If you work overtime, then you’ll be paid time-and-a-half

I’ll wash the car if the weather is nice

If 2 divides evenly into x, then x is an even number

Definitions

• Statement 4

–

• Statement 5

–

I’ll be a millionaire when I win monopoly

All equiangular triangles are equilateral

Helpful Hint

• Always re-write the statements into formIf-Then

Definitions

• Symbolic Notation

– Using letters to represent each

– Usually done as

– p:

– q:

statement

p -> q

hypothesis

conclusion

Examples

• Example 1

Write each of the 5 statements in symbolic notation.

#1

p: you work overtime

q: you’ll be paid time-and-a-half

p -> q

#2

p: the weather is nice

q: I’ll wash the car

p -> q

#3

p: 2 divides evenly into x

q: x is an even number

p -> q

#4

p: I win monopoly

q: I’ll be a millionaire

p -> q

Examples

• Example 1

#5

p: a triangle is equiangular

q: it is equilateral

p -> q

Definitions

• Negation

– The of a statement

– Written symbolically as

opposite

~

Examples

• Example 2

Using statement 1, write the negation of the p and q.

p: you work overtime

q: you’ll get paid time-and-a-half

~p: you don’t work overtime

~q: you don’t get paid time-and-a-half

Definitions

• Converse

– The of an if-then statement

– Symbolically:

• Inverse

– The of an if-then statement

– Symbolically:

opposite

q -> p

negation

~p -> ~q

Definitions

• Contrapositive

– The and of an if-then statement

– Symbolically:

– to the original statement

opposite negation

~q -> ~p

Logically equivalent

Examples

• Example 3

Use the statement: If n > 2, then n^2 > 4. Find the converse, inverse, and contrapositive. Determine if

each of them are true or false. If false, find a counterexample.

Converse

If n^2 > 4, then n > 2

Inverse

If n < 2, then n^2 < 4

Contrapositive

If n^2 < 4, then n < 2

False, n = -3 False, n = -3 True

Examples

• Example 4

Use the statement: If I am at Disneyland, then I am in California. Find the converse, inverse, and

contrapositive. Determine if each of them are true or false. If false, find a counterexample.

Converse

If I am in California, then I am at Disneyland

Inverse

If I am not in Disneyland, then I am not in California

Contrapositive

If I am not in California, then I am not in Disneyland

False, I could be in San Diego False, I could be

in San DiegoTrue

Note

• In example 4, we could use the same counterexample for the Converse and Inverse

– That is because the Converse and Inverse are

Logically Equivalent

Definitions

• Biconditional

– When a conditional statement and its converse are both

– In other words, p -> q and q -> p are both

– A biconditional is written like this:

– A biconditional is read as “p if and only if q”

True

true

p <-> q

Examples

• Example 5

Rewrite this statement as a biconditional statement.

If two points are on the same line, then they are collinear.

Two points are on the same line if and only if they are collinear.

• “If and only if” can be abbreviated as iff

Examples

• Example 6

The following is a true statement. Determine the true statements within this biconditional.

m∠ABC > 90° if and only if ∠ABC is an obtuse angle.

If ∠ABC > 90°,

then ∠ABC is

an obtuse angle

If ∠ABC is an

obtuse angle, then

m∠ABC > 90°

• All geometric definitions are Biconditionals

Examples

• Example 7

p: x > 10 q: 2x < 50

a. Is p -> q true? If not, find a counterexample.

b. Is q -> p true? If not, find a counterexample.

c. Is ~p -> ~q true? If not, find a counterexample.

d. Is ~q -> ~p true? If not, find a counterexample.

true

False, x = 15

False, x = 15True

• HW #22: #31-40, 43-46, 48-50, 52, 55