Conditional Statements in Mathematics
A statement in the form “If _____, then _______.”
is called a
Conditional Statement
A conditional statement has two parts:
hypothesis
conclusion
If ______________ , then _______________ .
Ex1: If today is Monday, then
yesterday was Sunday.
Hypothesis: Today is Monday.
Conclusion: Yesterday was
Sunday.
Examples: Identify the Hypothesis and Conclusion of
each conditional statement.
Examples: Identify the Hypothesis and Conclusion of
each conditional statement.
Ex2. If x = 5, then 2x = 10.
Ex3. If x = 6, then x
2
= 36.
Hypothesis: x = 5
Hypothesis: x = 6
Conclusion: 2x = 10.
Conclusion: x
2
= 36.
Rewriting conditional statements into
the “if…then…” format.
Ex4) 3x + 2 = -13 implies that x = -5.
Rewrite as “If 3x + 2 = -13, then x = -5.”
Hypothesis: 3x + 2 = -13
Conclusion: x = -5
Ex5) A number is divisible by 2 if it
is divisible by 6.
Rewrite as “If a number is divisible by 6,
then it is divisible by 2.”
Hypothesis: A number is divisible by 6.
Conclusion: The number is divisible by 2.
To find the
converse
of a conditional
statement, you switch the hypothesis
and conclusion.
Ex6: If today is Monday, then yesterday was Sunday.
Hypothesis: Today is Monday.
Conclusion: Yesterday was Sunday.
Converse: If yesterday was Sunday, then today is
Monday.
Not all converses will be true!!
Ex7. If you live in Doylestown, then you live in
Pennsylvania.
Hypothesis:
You live in Doylestown.
Conclusion:
You live in Pennsylvania.
Converse: If you live in Pennsylvania, then you
live in Doylestown.
Is this converse true?
No.
In order to prove the converse is false,
you
must
provide a counterexample.
A
counterexample
is an example for
which the hypothesis is true, but the
conclusion is false.
Counterexample:
You live in Chalfont.
Example: For the conditional statement,
a)
identify the hypothesis and conclusion,
b)
write the converse of the statement and determine if
the converse is true or false.
c) If the converse is false, provide a counterexample.
a) Hypothesis: x = -5
Conclusion:
b) Converse:
False
c) Counterexample: x = 5.
Biconditional Statements
use the phrase “if and only if” to indicate
when a conditional
and
its converse are both
true.
If today is Monday,
then yesterday was Sunday.
Rewrite
: Today is Monday
if and only if
today yesterday was Sunday.
You can rewrite definitions as biconditionals.
Ex2.
Congruent segments
are segments
that have equal lengths.
Ex1.
Coplanar points
are points all in one
plane.
Points are coplanar if and only if they
all lie in one plane.
Segments are congruent if and only
if their lengths are equal.
Ex3.
Complementary angles
are two angles
whose measures add up to 90°.
Two angles are complementary if and
only if their measures add up to 90°.
A conditional statement in the form "If ___, then ___" comprises a hypothesis and a conclusion. Examples and explanations on identifying hypotheses and conclusions, rewriting statements, finding converses, and providing counterexamples are provided in this informative content snippet.
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Presentation Transcript
A statement in the form If _____, then _______. is called a Conditional Statement A conditional statement has two parts: If ______________ , then _______________ . hypothesis conclusion
Examples: Identify the Hypothesis and Conclusion of each conditional statement. Ex1: If today is Monday, then yesterday was Sunday. Hypothesis: Today is Monday. Conclusion: Yesterday was Sunday.
Examples: Identify the Hypothesis and Conclusion of each conditional statement. Ex2. If x = 5, then 2x = 10. Hypothesis: x = 5 Conclusion: 2x = 10. Ex3. If x = 6, then x2 = 36. Hypothesis: x = 6 Conclusion: x2 = 36.
Rewriting conditional statements into the if then format. Ex4) 3x + 2 = -13 implies that x = -5. Rewrite as If 3x + 2 = -13, then x = -5. Hypothesis: 3x + 2 = -13 Conclusion: x = -5
Ex5) A number is divisible by 2 if it is divisible by 6. Rewrite as If a number is divisible by 6, then it is divisible by 2. Hypothesis: A number is divisible by 6. Conclusion: The number is divisible by 2.
To find the converse of a conditional statement, you switch the hypothesis and conclusion. Ex6: If today is Monday, then yesterday was Sunday. Hypothesis: Today is Monday. Conclusion: Yesterday was Sunday. Converse: If yesterday was Sunday, then today is Monday.
Not all converses will be true!! Ex7. If you live in Doylestown, then you live in Pennsylvania. Hypothesis: Conclusion: You live in Doylestown. You live in Pennsylvania. Converse: If you live in Pennsylvania, then you live in Doylestown. Is this converse true? No.
In order to prove the converse is false, you must provide a counterexample. A counterexample is an example for which the hypothesis is true, but the conclusion is false. Counterexample: You live in Chalfont.
Example: For the conditional statement, a) identify the hypothesis and conclusion, b) write the converse of the statement and determine if the converse is true or false. c) If the converse is false, provide a counterexample. If x , 5 then x . 5 Ex8) = = x = 5 a) Hypothesis: x = -5 Conclusion: If x , 5 then x . 5 b) Converse: False = = c) Counterexample: x = 5.
Biconditional Statements use the phrase if and only if to indicate when a conditional and its converse are both true. If today is Monday, then yesterday was Sunday. Rewrite: Today is Monday if and only if today yesterday was Sunday.
You can rewrite definitions as biconditionals. Ex1. Coplanar points are points all in one plane. Points are coplanar if and only if they all lie in one plane. Ex2. Congruent segments are segments that have equal lengths. Segments are congruent if and only if their lengths are equal.
Ex3. Complementary angles are two angles whose measures add up to 90 . Two angles are complementary if and only if their measures add up to 90 .
