     # 1.3.2 Conditional Statements

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• Conditional Statements

The student is able to (I can):

Identify, write, and analyze conditional statements.

Write the inverse, converse, and contrapositive of a conditional statement.

Write a counterexample to a false conjecture.

• conditional statement

hypothesis

conclusion

A statement that can be written as an if-then statement.

Example: IfIfIfIf today is Saturday, thenthenthenthen we dont have to go to school.

The part of the conditional following the word if (underline once).

today is Saturday is the hypothesis.

The part of the conditional following the word then (underline twice).

we dont have to go to school is the conclusion.

• Notation

Examples

Conditional statement: p q, where

p is the hypothesis and

q is the conclusion.

Identify the hypothesis and conclusion:

1. If I want to buy a book, then I need some money.

2. If today is Thursday, then tomorrow is Friday.

3. Call your parents if you are running late.

• Notation

Examples

Conditional statement: p q, where

p is the hypothesis and

q is the conclusion.

Identify the hypothesis and conclusion:

1. If I want to buy a book, then I need some money.

2. If today is Thursday, then tomorrow is Friday.

3. Call your parents if you are running late.

• Examples

To write a statement as a conditional, identify the sentences hypothesis and conclusion by figuring out which part of the statement depends on the other.

Write a conditional statement:

Two angles that are complementary are acute.

Even numbers are divisible by 2.

• Examples

To write a statement as a conditional, identify the sentences hypothesis and conclusion by figuring out which part of the statement depends on the other.

Write a conditional statement:

Two angles that are complementary are acute.

If two angles are complementary, then they are acute.

Even numbers are divisible by 2.

If a number is even, then it is divisible by 2.

• As we saw with inductive reasoning, to prove a conjecture false, you just have to come up with a counterexample.

The hypothesis must be the same as the conjectures and the conclusion is different.

Example: Write a counterexample to the statement, If a quadrilateral has four right angles, then it is a square.

• As we saw with inductive reasoning, to prove a conjecture false, you just have to come up with a counterexample.

The hypothesis must be the same as the conjectures and the conclusion is different.

Example: Write a counterexample to the statement, If a quadrilateral has four right angles, then it is a square.

A counterexample would be a quadrilateral that has four right angles (true hypothesis) but is not a square (different conclusion). So a rectanglerectanglerectanglerectangle would work.

• Each of the conjectures is false. What would be a counterexample?

If I get presents, then today is my birthday.

If Lamar is playing football tonight, then today is Friday.

• Each of the conjectures is false. What would be a counterexample?

If I get presents, then today is my birthday.

A counterexample would be a day that I get presents (true hyp.) that isnt my birthday (different conc.), such as Christmas.

If Lamar is playing football tonight, then today is Friday.

Lamar plays football (true hyp.) on days other than Friday (diff. conc.), such as games on Thursday.

• Examples Determine if each conditional is true. If false, give a counterexample.

1. If your zip code is 76012, then you live in Texas.

TrueTrueTrueTrue

2. If a month has 28 days, then it is February.

September also has 28 days, which proves the conditional false.

Texas76012

• negation of p Not p

Notation: ~p

Example: The negation of the statement Blue is my favorite color, is Blue is notnotnotnotmy favorite color.

Related Conditionals Symbols

Conditional p q

Converse q p

Inverse ~p ~q

Contrapositive ~q ~p

• Example Write the conditional, converse, inverse, and contrapositive of the statement:

A cat is an animal with four paws.

Type Statement

Conditional

(p q)

If an animal is a cat, then it has four paws.

Converse

(q p)

If an animal has four paws, then it is a cat.

Inverse

(~p ~q)

If an animal is not a cat, then it does not have four paws.

Contrapos-itive

(~q ~p)

If an animal does not have four paws, then it is not a cat.

• Example Write the conditional, converse, inverse, and contrapositive of the statement:

When n2 = 144, n = 12.

Type StatementTruth Value

Conditional

(p q)If n2 = 144, then n = 12.

F

(n = 12)

Converse

(q p)If n = 12, then n2 = 144. T

Inverse

(~p ~q)If n2 144, then n 12 T

Contrapos-itive

(~q ~p)

If n 12, then n2 144F

(n = 12)

• biconditional A statement whose conditional and converse are both true. It is written as

p if and only if qp if and only if qp if and only if qp if and only if q, p iff qp iff qp iff qp iff q, or p p p p qqqq.

To write the conditional statement and converse within the biconditional, first identify the hypothesis and conclusion, then write p q and q p.

A solution is a base iff it has a pH greater than 7.

p q: If a solution is a base, then it has a pH greater than 7.

q p: If a solution has a pH greater than 7, then it is a base.

• Writing a biconditional statement:

1. Identify the hypothesis and conclusion.

2. Write the hypothesis, if and only if, and the conclusion.

Example: Write the converse and biconditional from:

If 4x + 3 = 11, then x = 2.

Converse: If x = 2, then 4x + 3 = 11.

Biconditional: 4x + 3 = 11 iff x = 2.

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