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 conditional conditional conditional statement statement statement statement – a statement that can be written as

an “if-then” statement.

Example: IfIfIfIf today is Saturday, thenthenthenthen we don’t have to go to

school.

hypothesis hypothesis hypothesis hypothesis – the part of the conditional followingfollowingfollowingfollowing the word

“if” (underline once).

“today is Saturday” is the hypothesis.

conclusion conclusion conclusion conclusion – the part of the conditional followingfollowingfollowingfollowing the word

“then” (underline twice).

“we don’t have to go to school” is the conclusion.

Examples

NotationNotationNotationNotation

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

NotationNotationNotationNotation

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 sentence’s 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 sentence’s 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.

To prove a conjecture false, you just have to come up with a

counterexample.

• The hypothesis must be the samesamesamesame as the conjecture’s and

the conclusion is differentdifferentdifferentdifferent.

Example: Write a counterexample to the statement, “If a

quadrilateral has four right angles, then it is a square.”

To prove a conjecture false, you just have to come up with a

counterexample.

• The hypothesis must be the samesamesamesame as the conjecture’s and

the conclusion is differentdifferentdifferentdifferent.

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.

Examples 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.

Examples 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 isn’t 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.

Texas

76012

negation of negation of negation of negation of pppp – “Not p”

Notation: ~p

Example: The negation of the statement “Blue is my favorite

color,” is “Blue is notnotnotnot my favorite color.”

Related ConditionalsRelated ConditionalsRelated ConditionalsRelated Conditionals SymbolsSymbolsSymbolsSymbols

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.”

TypeTypeTypeType StatementStatementStatementStatement

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.

Contrapositive

(~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.”

TypeTypeTypeType StatementStatementStatementStatement Truth ValueTruth ValueTruth ValueTruth 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

Contrapositive

(~q→ ~p)If n ≠ 12, then n2 ≠ 144

F

(n = –12)

biconditional biconditional biconditional biconditional – a statement whose conditional and converse

are both true. It is written as

“pppp if and only if if and only if if and only if if and only if qqqq”, “pppp iff iff iff iff qqqq”, or “pppp↔↔↔↔ 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.