Identify, write, and analyze conditional statements. Write the converse, inverse, and contrapositive of a conditional statement. Write biconditional statements.

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. conditional statement – a statement that can be written as
an “if-then” statement.
Example: If today is Saturday, then we don’t have to
go to school.
hypothesis – the part of the conditional following the word
“if” (underline once).
“today is Saturday” is the hypothesis.
conclusion – the part of the conditional following the word
“then” (underline twice).
“we don’t have to go to school” is the conclusion.

3. Notation
Conditional statement: p  q, where
p is the hypothesis and q is the conclusion.
Examples
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. Notation
Conditional statement: p  q, where
p is the hypothesis and q is the conclusion.
Examples
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. 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.
Examples
Write a conditional statement:
• Two angles that are complementary are acute.
• Even numbers are divisible by 2.

6. 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.
Examples
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 same as the conjecture’s and
the conclusion is different.
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 same as the conjecture’s 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 rectangle would work.

9. 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. 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.
True
2. If a month has 28 days, then it is
February.
September also has 28 days, which
proves the conditional false.

12. negation of p – “Not p”
Notation: ~p
Example: The negation of the statement “Blue is my favorite
color,” is “Blue is not my favorite color.”





13. Write the conditional, converse, inverse, and contrapositive
of the statement:
“A cat is an animal with retractable claws.”

14. Write the conditional, converse, inverse, and contrapositive
of the statement:
“A cat is an animal with retractable claws.”
Type Statement
Conditional
(p  q)
If an animal is a cat, then it has retractable
claws.
Converse
(q  p)
If an animal has retractable claws, then it is
a cat.
Inverse
(~p  ~q)
If an animal is not a cat, then it does not
have retractable claws.
Contrapos-itive
(~q  ~p)
If an animal does not have retractable
claws, then it is not a cat.

15. Write the conditional, converse, inverse, and contrapositive
of the statement:
“When n2 = 144, n = 12.”
Type Statement Truth 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  144
F
(n = –12)

16. biconditional – a statement whose conditional and converse
are both true. It is written as
“p if and only if q”, “p iff q”, or “p  q”.
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.

17. 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.