Honors Math 2 and Cambridge Math 2 Project
For your summer math project, you will be looking at an introduction to logic.
Before you complete the project, please look over the PowerPoint slides in this
packet.
If you have questions, please feel free to email Mr. Brust or Mrs. Kerns. We may
not check our email every day, but we will respond within 2-3 days. This
assignment is due the first day of classes in August.
Have a great summer!!
Sincerely,
Mr. Patrick Brust
[email protected]
Mrs. Jody Kerns
[email protected]
Conditional
Statements
Lesson 2-1 Conditional
Statements
1
Conditional Statement
Definition: A conditional statement is a statement that
can be written in if-then form.
“If _____________, then ______________.”
“If p, then q.”
Example:
If the animal is a zebra, then it has stripes.
Continued……
Lesson 2-1 Conditional
Statements
2
Conditional Statement - continued
Conditional Statements have two parts:
The hypothesis is the part of a conditional statement that follows
“if” (when written in if-then form.) This is often represented by
the letter “p.”
The conclusion is the part of an if-then statement that follows
“then” (when written in if-then form.) This is often represented
by the letter “q.”
Lesson 2-1 Conditional
Statements
3
Writing Conditional Statements
Conditional statements can be written in “if-then” form to
emphasize which part is the hypothesis and which is the
conclusion.
Example: Vertical angles are congruent. can be written as...
Conditional
Statement: If two angles are vertical, then they are congruent.
Lesson 2-1 Conditional
Statements
4
Conditional Statements can be true or false:

A conditional statement is false only when the hypothesis is true,
but the conclusion is false.
A counterexample
is an example used to show that a
statement is not always true and therefore false.
If you live in North Carolina, then you live in
Raleigh.
Yes !!!
Is there a counterexample?
Statement:
Counterexample: I live in North Carolina, BUT I live in
Charlotte.
Lesson 2-1 Conditional
Statements
5
Symbolic Logic - continued
pq
~
is used to represent
is used to represent the word
Example 1:
p: the angle is obtuse
~p:
The angle is not obtuse
Lesson 2-1 Conditional
Statements
if p, then q
or
p implies q
“not”
6
Symbolic Logic - continued

is used to represent the word
Example:
“therefore”
Therefore, the statement is false.
 the statement is false
Lesson 2-1 Conditional
Statements
7
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.
THE CONVERSE OF A CONDITIONAL STATEMENT
IS NOT ALWAYS TRUE!!!!!!!
Lesson 2-1 Conditional
Statements
8
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.
Lesson 2-1 Conditional
Statements
9
Forms of Conditional Statements

Contrapositives are logically equivalent to the
original conditional statement.

If pq is true, then qp is true.

If pq is false, then qp is false.
Lesson 2-1 Conditional
Statements
10
Forms of Conditional Statements
Inverse: Negate both the hypothesis and
the 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
Lesson 2-1 Conditional
Statements
11
Forms of Conditional
Statements

Converse and inverse are logically
equivalent to one another.

If qp is true, then qp is true.

If qp is false, then qp is false.
Lesson 2-1 Conditional
Statements
12
Biconditional

When a conditional statement and its converse are both true,
the two statements may be combined.

Use the phrase if and only if (sometimes abbreviated: iff)
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.
Lesson 2-1 Conditional
Statements
13
Name: ________________________________ Block: _____ Date: _______________________
Honors Math 2 and Cambridge Math 2 Summer Project (Due: First Day of School)
A. Read the PowerPoint slides before starting your summer project.
Identify the hypothesis and the conclusion for each of the following conditional statements:
1.
If Colleen studies for her test, then she will pass.
Hypothesis: __________________________
2.
If Jose speeds on his motorcycle, then he will get a traffic ticket.
Hypothesis: __________________________
3.
Conclusion: __________________________
A triangle with two equal angles is isosceles.
Hypothesis: __________________________
4.
Conclusion: __________________________
Conclusion: __________________________
If the animal is a zebra, then it has stripes.
Hypothesis: __________________________
Conclusion: _________________________
5. Go back and place a circle around the “p” in each sentence and an X over the “q”. Then, highlight or
underline the “given” in each sentence. Leave the “prove” alone.
B. You have been provided 24 phrases that will be used to form three true conditional statements, their
inverses, their converses, and their contrapositives. Your task is to cut out each piece and arrange 12
sentences using those pieces. Your pieces will fill in the blanks for the if _____ then _____
statements. No handwritten copies will be accepted - please only cut and paste. Circle T for true and
F for false for each statement.
Sentence #1
T or F Conditional (true statement in if-then form):
If
____________________________________
Then _______________________________
T or F Converse:
If
____________________________________
Then _______________________________
T or F Inverse:
If
____________________________________
Then _______________________________
T or F Contrapositive:
If
____________________________________
Then _______________________________
Sentence #2
T or F Conditional (true statement in if-then form):
If
____________________________________
Then _______________________________
T or F Converse:
If
____________________________________
Then _______________________________
T or F Inverse:
If
____________________________________
Then _______________________________
T or F Contrapositive:
If
____________________________________
Then _______________________________
Sentence #3
T or F Conditional (true statement in if-then form):
If
____________________________________
Then _______________________________
T or F Converse:
If
____________________________________
Then _______________________________
T or F Inverse:
If
____________________________________
Then _______________________________
T or F Contrapositive:
If
____________________________________
Then _______________________________
it is a zebra
today is Monday
it is a square
it is a rectangle
it has stripes
it is not a square
it is not a zebra
it does not have stripes
tomorrow is Tuesday
it is not a rectangle
today is not Monday
tomorrow is not Tuesday
it is a zebra
it has stripes
it is a square
tomorrow is not Tuesday
today is Monday
it is not a zebra
tomorrow is Tuesday
it does not have stripes
it is not a square
it is not a rectangle
today is not Monday
it is a rectangle
C.
Disprove the following by providing one or more counterexamples that show the original statement is
false. You do NOT have to write in complete sentences. Be careful, some of them are actually true!! If it
is an absolutely true statement, write “true.”
5.
Every day of the week has an "R" in it.
6.
If it’s a planet, then it’s smaller than the Earth.
7.
If it’s a vegetable, then it is not orange.
8.
All polygons are quadrilaterals.
9.
If I pass the class, then I earned an A.
10.
All triangles are isosceles.
11.
All integers are even or odd.
12. If your clothes are wet, then it is raining.
13. If a number is divisible by 2, then it is a multiple of 4.
14. If angle A is acute, then m< 𝐴 = 35°.
15. If you are in North Carolina, then you are in Charlotte.
16. If a candidate wins the popular vote, then the candidate is elected President of the United States.
17. 𝑥 2 will always be greater than 𝑥
18. If a snake is poisonous, then it has a triangular head.
19. If a bill passes the House and Senate, then it becomes law.
20. If the governor of North Carolina is a Democrat, then the lieutenant governor of North Carolina is a
Democrat.
21. If a number is prime, then that number must also be odd.
22. If it is the weekend, then today is Saturday.
23. If (x,y)
( y , -x) then the transformation was 270 degrees counterclockwise.
24. If 2/3 of the states ratify an amendment, then the amendment is added to the Constitution.
25. If it is an angle greater than 90 then it is obtuse.
26. If I were rich, then I’d be happy.
27. If 𝑥 2 = 49 then x = 7.
28. If 51% of the Senate votes for cloture, then the filibuster is broken.
29. If you are 16 years old, then you have a driver’s license.
30. If you are 35 years old, then you can be President of the United States.
D. Create the following – 1) Three absolutely TRUE conditional statements. 2) Three false conditionals that
may challenge your peers. (Include your counterexamples).
E.
Create your own unique conditional statements that satisfy the charts’ truth values. You cannot re-use
any statements that have already been used in other parts of this project.
conditional
1.
T
2.
T
3.
F
4.
F
5.
T
6.
T
1.
Converse
T
F
T
F
T
Inverse
T
F
T
F
F
Contrapositive
T
T
F
F
T
F
2.
3.
4.
5.
6.______________________________________________________________________
F.
In general, of the four types of statements (conditional, converse, inverse, and contrapositive), which are
“logically equivalent”?