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2.1 ESSENTIAL QUESTION
When is a conditional statement true or false?
September 15, 2015 2.1 CONDITIONAL STATEMENTS
WHAT YOU WILL LEARN
oWrite conditional statements.
oUse definitions written as conditional statements.
oWrite biconditional statements.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
CONDITIONAL
A type of logical statement that has two parts, a hypothesis and a conclusion.
A conditional can be written in IF-THEN form.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
SHORTHAND
If HYPOTHESIS, then CONCLUSION.
If P, then Q.
In the study of logic, P’s and Q’s are universally accepted to represent hypothesis and conclusion.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EXAMPLE 1
If I study hard, then I will get good grades.
HYPOTHESIS
I study hard
CONCLUSION
I will get good grades.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
CAN YOU IDENTIFY THE HYPOTHESIS AND CONCLUSION?
If today is Monday, then tomorrow is Tuesday.
Hypothesis: today is Monday
Conclusion: tomorrow is Tuesday.
Note: IF is NOT part of the hypothesis, and THEN is NOT part of the conclusion.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
YOUR TURN
Underline the hypothesis and circle the conclusion.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
1. If the weather is warm, then we should go swimming.
2. If you want good service, then take your car to Joe’s Service Center.
REWRITING STATEMENTS.
oUse common sense.
oDon’t over analyze it.
oMake sure the sentence is grammatically correct.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
The hypothesis always follows “IF.”
No “if?” The first part is usually the hypothesis.
Make your English teacher proud!Does it sound right?
EXAMPLE 2A
Rewrite the following statement in if-then form:
All birds have feathers.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
What is the hypothesis?
What is the conclusion? have feathers
All birds
If-then form?
If an animal is a bird, then it has feathers.
EXAMPLE 2B
Rewrite the following statement in if-then form:
You are in Texas if you are in Houston.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
What is the hypothesis?
What is the conclusion? You are in Texas
You are in Houston
If-then form?
If you are in Houston, then you are in
Texas.
EXAMPLE 2C
Rewrite the following statement in if-then form:
An even number is divisible by 2.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
What is the hypothesis?
What is the conclusion? Divisible by 2.
An even number
If-then form?
If a number is even, then it is divisible by 2.
YOUR TURN
Rewrite the conditional statement in if-then form.
If yesterday was Sunday, then today is Monday.
If an object is one foot long, then it measures 12 inches.
or
If an object measures 12 inches, then it is one foot long.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
3. Today is Monday if yesterday was Sunday.
4. An object that measures 12 inches is one foot long.
NEGATION
The negative of the original statement. Examples:
I am happy.
I am not happy.
mC = 30°.
mC 30°.
A, B and C are on the same line.
A, B and C are not on the same line.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EXAMPLE 3
Write the negation of each statement.
a. The ball is red.
The ball is not red.
b. The cat is not black.
The cat is black.
c. The car is white.
The car is not white.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
RELATED CONDITIONAL STATEMENTS
Looking at the conditional statement: If p, then q.
There are three similar statements we can make.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
o Converseo Inverseo Contrapositive
CONVERSE
The converse of a statement is formed by
switching the hypothesis and the conclusion.
If you play drums, then you are in the band.
Conditional:
Converse:
If you are in the band, then you play drums.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If Q, then P.
EXAMPLE 4
Write the converse of the statement below.
Answer:
If you play on the tennis team, then you like tennis.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If you like tennis, then you play on the tennis team.
INVERSE
The inverse is made by taking the negation of
the hypothesis and of the conclusion.
Conditional:
If x = 3, then 2x = 6.
Inverse:
If x 3, then 2x 6.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If not P, then not Q.
EXAMPLE 5
Write the inverse of the statement below.
Answer:
If today is not Monday, then tomorrow is not Tuesday.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If today is Monday, then tomorrow is Tuesday.
CONTRAPOSITIVE
The contrapositive is made by taking the inverse of the
converse, or, the converse of the inverse.
Conditional:
If I am in 10th grade, then I am a sophomore.
Contrapositive:
If I am not a sophomore, then I am not in 10th
grade.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If not Q, then not P.
EXAMPLE 6
Write the contrapositive of the statement below.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If x is odd, then x + 1 is even.
x + 1 is not evenNegateNegate
x is not odd
If x+1 is not even, then x is not odd.
LOGICAL STATEMENTS
If I live in Mesa, then I live in Arizona.
Converse: (switch hypothesis and conclusion)
If I live in Arizona, then I live in Mesa.
Inverse: (negate hypothesis and conclusion)
If I don’t live in Mesa, then I don’t live in Arizona.
Contrapositive: (switch and negate both)
If I don’t live in Arizona, then I don’t live in Mesa.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
YOUR TURN. WRITE THE CONVERSE, INVERSE, AND CONTRAPOSITIVE.
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA 20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA 20.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
REVIEW: LOGICAL STATEMENTS
September 15, 2015 2.1 CONDITIONAL STATEMENTS
Conditional: If P, then Q.
Converse: If Q, then P.
Inverse: If not P, then not Q.
Contrapositive: If not Q, then not P.
DEFINITION: PERPENDICULAR LINES
September 15, 2015 2.1 CONDITIONAL STATEMENTS
Two lines that intersect to form a right angle.
m
n
Notation:
m n
USING DEFINITIONS
You can write a definition as a conditional statement in if-then form. Let’s look at an example:
The conditional statement would be:
The converse statement also ends up being true:
September 15, 2015 2.1 CONDITIONAL STATEMENTS
Perpendicular Lines: two lines that intersect to
form a right angle.
If two lines intersect to form a right angle,
then they are perpendicular lines.
If two lines are perpendicular, then they
intersect to form a right angle.
BICONDITIONALS
September 15, 2015 2.1 CONDITIONAL STATEMENTS
If 2 s are complementary, then their sum is 90°. True
Converse
If the sum of 2 s is 90°, then they are complementary. True
When a conditional statement and its converse are both TRUE,
they can be written as a single biconditional statement. Let’s look
at an example:
Conditional
Biconditional
2 s are complementary if and only if their sum is 90°.
BICONDITIONALS (Continued)
September 15, 2015 2.1 CONDITIONAL STATEMENTS
Written with p’s and q’s,
a biconditional looks like this:
p if and only if q.
p iff q. or
Iff means “if and only if”.
MIND YOUR PS AND QS.
September 15, 2015 2.3 DEDUCTIVE REASONING 37
Conditional: If HYPOTHESIS, then CONCLUSION.
Let P represent the HYPOTHESIS.
Let Q represent the CONCLUSION.
Then the conditional is: If P, then Q.
The notation is: P Q.The symbol “” is often read as “implies”.
LOGICAL STATEMENTS
September 15, 2015 2.3 DEDUCTIVE REASONING 38
Conditional: P Q
Converse: Q P
Biconditional: P Q
EXAMPLE 7
September 15, 2015 2.3 DEDUCTIVE REASONING 39
Let P be the statement: “x = 3”
Let Q be the statement: “2x = 6”
Write:
P Q
Q P
P Q
If x = 3, then 2x = 6.
If 2x = 6, then x = 3.
x = 3 if and only if 2x = 6.
or 2x = 6 iff x = 3.
NEGATION
September 15, 2015 2.3 DEDUCTIVE REASONING 40
Use the symbol ~. Read it as “not”.
P is the statement “I like ice cream”
~P is read “Not P”
~P is the statement “I don’t like ice cream”
LOGICAL STATEMENTS – SYMBOLIC FORM
September 15, 2015 2.3 DEDUCTIVE REASONING 41
Conditional: P Q
Converse: Q P
Biconditional: P Q
Inverse: ~P ~Q
Contrapositive: ~Q ~P
EXAMPLE 8
September 15, 2015 2.3 DEDUCTIVE REASONING 42
P: it is summer
Q: it is hot
~P: It is not summer.
~P ~Q: If it is not summer, then it is not hot.
Q P: If it is hot, then it is summer.
YOUR TURN. WRITE YOUR ANSWERS DOWN ON YOUR PAPER.
September 15, 2015 2.3 DEDUCTIVE REASONING 43
P: I work hard
Q: I will get into college
1. What is P Q?
2. What is ~Q ~P?
3. Write ~P ~Q.
If I work hard, then I will get into college.
If I don’t get into college, then I didn’t work hard.
If I don’t work hard, then I won’t get into college.
TRUTH VALUES
•A conditional is either True or False.
•To show that it is true, you must have an argument (a proof) that it is true in all cases.
•To show that it is false, you need to provide at least one counterexample.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EXAMPLE 8True or false? If false provide a counter example.
If x2= 9, then x = 3.
FALSE!Counterexample: x could be –3.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EXAMPLE 9
If x = 10, then x + 4 = 14.
True! Proof:
x = 10
x + 4 = 10 + 4
x + 4 = 14
September 15, 2015 2.1 CONDITIONAL STATEMENTS
LET’S REVIEW
If today is Sunday, then we have school tomorrow.
A. If we have school tomorrow, then today is Sunday.
B. If we don’t have school tomorrow, then today is not Sunday.
C. If today is not Sunday, then we do not have school tomorrow.
Which of these is the Converse, Inverse, and Contrapositive? And what are the truth values?
Converse
Contrapositive
Inverse
False
False
False
False
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EQUIVALENT STATEMENTS
When two statements are both true or both false, they are called equivalent statements.
A conditional statement is always equivalent to its contrapositive.
The inverse and converse are also equivalent.
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EQUIVALENT STATEMENTS
Original:
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA 20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA 20.
TRUE
False
False
TRUE
September 15, 2015 2.1 CONDITIONAL STATEMENTS
EXAMPLE 10
Statement: If x = 5, then x2 = 25. TRUE
Contrapositive: If x2 25, then x 5. TRUE
Converse: If x2 = 25, then x = 5. FALSE – could be –5.
Inverse: If x 5, then x2 25. FALSE
September 15, 2015 2.1 CONDITIONAL STATEMENTS
WHY IS THIS IMPORTANT?
Geometry is stated in rules of logic.
We use logic to prove things.
It teaches us to think clearly and without error.
It impresses girl friends (or boy friends).
You can talk like…
September 15, 2015 2.1 CONDITIONAL STATEMENTS