Download Conditional Statements

Conditional Statements
Mrs. Spitz
Modifyied by Mrs. Ortiz-Smith
Geometry
Objective

Recognize and analyze a conditional
statement
Conditional Statement

A logical statement has 2 parts:
hypothesis & conclusion

Can be written in “if-then” form
Conditional Statement

Hypothesis is the part after the word “If”

Conclusion is the part after the word
“then”
Ex: Underline the hypothesis &
circle the conclusion.
If you are a brunette, then you have brown hair.
hypothesis
conclusion
Ex: Rewrite the statement in “if-then” form
Vertical angles are congruent.
If there are 2 vertical angles, then they are
congruent.
If 2 angles are vertical, then they are
congruent.
Ex: Rewrite the statement in “if-then” form
An object weighs one ton if it weighs
2000 lbs.
If an object weighs 2000 lbs., then it weighs
one ton.
Counterexample
Is used to show that a conditional
statement is false.
Ex: Provide a counterexample.

If x2=81, then x must equal 9.
counterexample: x could be -9
because (-9)2=81, but x≠9.
Negation

Writing the opposite of a statement.
Ex: negate x=3
x≠3
 Ex: negate t>5
t 5

R U A GEN!US?
To Thales the primary question was not what do we know,
but how do we know it. ~ Aristotle
Symbolic Logic

Symbols can be used to connect statements.
Hypothesis is represented by “p”
Conclusion is represented by “q”
if p, then q
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Symbolic Logic
pq
represents
if p, then q
Example:
p: a number is prime
q: a number has exactly two divisors
pq: If a number is prime, then it has
exactly two divisors.
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Symbolic Logic
~
Example:
represents the word not
p: the angle is obtuse
~p: The angle is not obtuse
~p means that the angle could be
acute, right, or straight.
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Symbolic Logic
Example:
p: I am not happy
~p: I am happy
~p took the “not” out- it would have
been a double negative (not not)
Forms of Conditional Statements
Converse

Switch the hypothesis & conclusion.

Ex: “If you are a brunette, then you
have brown hair.”
If you have brown hair, then you are a
brunette.
Forms of Conditional Statements
Converse:
Switch the hypothesis and conclusion (q  p)
pq
If two angles are vertical,
then they are congruent.
qp
If two angles are congruent,
then they are vertical.
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Forms of Conditional Statements
Inverse

Negate the hypothesis & conclusion.

Ex: “If you are a brunette, then you
have brown hair.”
If you are not a brunette, then you do
not have brown hair.
Forms of Conditional Statements
Inverse:
State the opposite of both the hypothesis and conclusion.
(~p~q)
pq : If two angles are vertical,
then they are congruent.
~p~q: If two angles are not vertical,
then they are not congruent.
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Forms of Conditional Statements
Contrapositive
Negate, then switch the hypothesis &
conclusion.
 Ex: “If you are a brunette, then you
have brown hair.”

If you do not have brown hair, then you
are not a brunette.
Forms of Conditional Statements
Contrapositive:
Switch the hypothesis and conclusion and state their
opposites. (~q~p)
pq : If two angles are vertical, then they are
congruent.
~q~p: If two angles are not congruent, then they are
not vertical.
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The original conditional statement &
its contrapositive will always have
the same meaning.
The converse & inverse of a
conditional statement will always
have the same meaning.
Biconditional
 When
a conditional statement and its
converse are both true, they may be
combined using the phrase
if and only if (abbreviated: iff)
Conditional Statements
Biconditional
Example:
Statement: If an angle is right, then
it has a measure of 90.
Converse: If an angle measures 90,
then it is a right angle.
Biconditional: An angle is right if and
only if it measures 90.