* Identify, write, and analyze conditional statements * Write the inverse, converse, and contrapositive of a conditional statement * Write a counterexample to a false conjecture * Write a biconditional statement

1. 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.

2. conditionalconditionalconditionalconditional statementstatementstatementstatement – 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.
hypothesishypothesishypothesishypothesis – the part of the conditional followingfollowingfollowingfollowing the word
“if” (underline once).
“today is Saturday” is the hypothesis.
conclusionconclusionconclusionconclusion – the part of the conditional followingfollowingfollowingfollowing the word
“then” (underline twice).
“we don’t have to go to school” is the conclusion.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.”

8. 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.

9. 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.

10. 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.

11. 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

12. negation ofnegation ofnegation ofnegation 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

13. 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.

14. 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)

15. biconditionalbiconditionalbiconditionalbiconditional – a statement whose conditional and converse
are both true. It is written as
“pppp if and only ifif and only ifif and only ifif and only if qqqq”, “pppp iffiffiffiff 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.

16. 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.