Download Math 8, Unit 14C: Conditional Statements Notes Conditional

Math 8, Unit 14C: Conditional Statements
Notes
Conditional Statements
Objectives: (5.9) The student will represent logical relationships using conditional statements.
We often make if-then statements in our everyday conversation. For instance, “If I study for the
test, then I will get a good grade” or “If it is raining, then I will get wet” are examples of if-then
statements. In math, an if-then statement is called a conditional statement. The part that
follows the if is called the hypothesis, and the part that follows the then is the conclusion.
conditional statement
If I study for the test, then I will get a good grade.
hypothesis
conclusion
The hypothesis is often represented by the letter p. The conclusion is often shown as q.
Therefore, a conditional statement is often written as p → q , which is read “If p, then q”. Let’s
practice writing conditional statements.
Example:
Let p: There is a red sky at night.
Let q: Weather will be good the next day.
p → q : If there is a red sky at night, then weather will be good the next day.
Practice:
Let p: I get a job.
Let q: I earn money.
Let r: I buy concert tickets.
Let s: I spend all my money.
Use the legend above to “translate” the following conditional statements:
If I get a job, then I earn money.
1. p → q
2. q → r
If I earn money, then I buy concert tickets.
3. p → r
If I get a job, then I buy concert tickets.
4. r → s
If I buy concert tickets, then I spend all my money.
Let t: It is raining outside.
Let u: I will use an umbrella.
Let v: I will get wet.
Use the legend to translate the following from English to symbols.
1. If it is raining outside, I will get wet.
t→v
2. If it is raining outside, I will use an umbrella.
t→u
3. If I use an umbrella, it is raining outside.
u→t
Supplementary Materials
Math 8, Unit 14C: Conditional Statements
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Symbols of Logic
We know that “If p, then q” can be written as p → q . Other symbols used are:
“therefore” is written ∴
“not p” is written  p
The statements below illustrate translating from statement form to symbolic form.
Statement
Symbolic Form
If yesterday was Tuesday, then today is Wednesday.
Let p: Yesterday was Tuesday.
Let q: Today is Wednesday.
p→q
If yesterday was Tuesday, then today is not Thursday.
Let p: Yesterday was Tuesday.
Let q: Today is Thursday.
p → q
If I do not study, then I will fail the test.
Let p: I study.
Let q: I will fail the test.
 p→q
In the following exercises, shorten the statements by using the symbols of logic.
1. If p, then not q.
p → q
2. If not q, then not p
 q → p
3. If not r, then q.
r→q
4. If s, then p.
s→ p
Translate the symbolic statements into English statements.
Let p: The number is even.
Let q: The number is divisible by 2.
Let r: The number is odd.
5. If  q, then  p.
If the number is not divisible by 2, then the number is not even.
If the number is divisible by 2, then the number is not odd.
6. If q → r.
7. If q → p.
If the number is divisible by 2, then the number is even.
(Please note that not all conditional statements are necessarily true.)
8. p → r
If the number is even, then the number is odd.
9. q → p
If the number is divisible by 2, then the number is not even.
Supplementary Materials
Math 8, Unit 14C: Conditional Statements
Page 2 of 5
More Conditional Statements
Every conditional statement has three other conditional statements associated with it. If we write
the original statement as p → q , then
Converse :
q→ p
Inverse:
 p → q
Contrapositive:
 q → p
In logic, conditional statements can be either true or false. If a statement is true, is the converse
true or false? Inverse? Contrapositive? Try the following example:
Statement:
If it is a rose, then it is a flower.
p→q
true
Converse:
If it is a flower, then it is rose.
q→ p
false
Inverse:
If it is not a rose, then it is not a flower.
 p → q
false
Contrapositive: If it is not a flower, then it is not a rose.
 q → p
true
For the following statement, write the converse, inverse, and contrapositive. Then decide if each
is true or false.
Statement:
Converse:
Inverse:
If the point is represented by ( 3, −2 ) , then the
point is in quadrant IV.
If the point is in quadrant IV, then the point
is represented by ( 3, −2 ) .
If the point is not represented by ( 3, −2 ) , then
the point is not in quadrant IV.
Contrapositive: If the point is not in quadrant IV, then the point is
not represented by ( 3, −2 ) .
true
false
false
true
You can make the conjecture that if the statement is true, then only the ___________________
(converse, inverse, contrapositive) is always true.
(contrapositive)
Supplementary Materials
Math 8, Unit 14C: Conditional Statements
Page 3 of 5
Forms of Valid Reasoning
In the world of logic, there are many basic forms of valid reasoning. Two of the most common
are shown below:
p→q
p→q
p
q
∴q
∴ p
We also know the contrapositive is true.
p→q
 q → p
The arguments below are examples of arguments showing valid reasoning. They are shown in
both English and symbolic form.
English Argument
Symbolic Translation
If Darla had the flu, then Darla gave Marcus
the flu. Darla had the flu. Therefore, Darla gave
Marcus the flu.
Let p: Darla had the flu.
Let q: Darla gave Marcus the flu.
p→q
p
∴q
If 2 x + 5 =
11, then x = 3. But x ≠ 3 .
Therefore, 2 x + 5 ≠ 11
Let p: 2 x + 5 =
11
Let q: x = 3
p→q
q
∴ p
If two angles are vertical angles, then the two
angles are congruent. Therefore, if two angles are not
congruent, then they are not vertical angles.
Let p: Two angles are vertical angles.
Let q: The two angles are congruent.
p→q
∴ q → p
Now let’s practice.
Determine if the following statements do or do not fit the logically valid patterns. Set up a legend
to show what each letter represents; then translate the argument into symbolic form. If it is a
valid form of reasoning, identify a conclusion in English. If the statements do not fit a logically
valid reasoning pattern, write “no valid conclusion”.
Supplementary Materials
Math 8, Unit 14C: Conditional Statements
Page 4 of 5
Example: If Marty is a dog, then Marty can bark.
Marty cannot bark.
Conclusion: Therefore, Marty is not a dog.
1. If students are smart, then students will study.
Students are smart.
Conclusion: Therefore, students will study.
2. If you are open-minded, then you listen to both
sides of the story. You do not listen to both sides
of the story.
Conclusion: Therefore, you are not open-minded.
3. If the triangle is isosceles, then the base angles are
equal. The base angles are not equal.
Conclusion: Therefore, the triangle is not isosceles.
4. If you buy Snickers candy bars, you like
chocolate. You do not buy Snickers.
No valid conclusion
5. If the triangle is a right triangle, then the triangle
has a 90° angle. The triangle is a right triangle.
Conclusion: Therefore, the triangle has a
90° angle.
Supplementary Materials
Let p: Marty is a dog.
Let q: Marty can bark.
p→q
q
∴ p
Let p: Students are smart.
Let q: Students will study.
p→q
p
∴q
Let p: You are open-minded.
Let q: You listen to both sides of the story.
p→q
q
∴ p
Let p: The triangle is isosceles.
Let q: The base angles are equal.
p→q
q
∴ p
Let p: You buy Snickers candy bars.
Let
q: qYou like chocolate.
p→
p
not a valid form
Let p: The triangle is a right triangle.
Let q: The triangle has a 90° angle.
p→q
p
∴q
Math 8, Unit 14C: Conditional Statements
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