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Additional Topics for the
Mid-Term Exam
Propositions
• Definition: A proposition is a statement
with a truth value. That is, it is a statement
that is true or else a statement that is
false.
• Next are some examples with their truth
values.
Propositions
• Denver is the capital of Colorado. (true)
• The main campus of the University of Kentucky is
in Athens, Ohio. (false)
• It will rain in Knoxville tomorrow. (We
cannot know the truth of this statement
today, but it is certainly either true or
false.)
• 2+2=4 (true)
• 2+2=19 (false)
• There are only finitely many prime
numbers. (false)
Propositions
• On the other hand, here are some examples of
expressions that are not propositions.
Not Propositions
• Where is Denver? (This is neither true nor false. It
is a question.)
• Find Denver! (This is neither true nor false. It is a
command.)
• Blue is the best color to paint a house. (This is a
matter of opinion, not truth. It may be your favorite
color, but it is not objective truth.)
• Not Propositions
– Questions
– Command
– Opinion
Propositions
• Notation: represented by a lower case letter,
usually p, q & r.
• Example: designating a proposition;
p: the glass is full.
q: I am 35 years old.
r: today is Monday.
Negation
• Negation (not): The negation of a proposition
p is “not p”.
• We denote it ~p.
• It is true if p is false and vice versa.
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Negation
• Formed by inserting “not” or “do not” in the
proposition.
• Example;
p: Harvard is a college.
~p: Harvard is not a college.
q: I am 35 years old.
~q: I am not 35 years old.
Conditional
• A conditional is a compound proposition of the form
“if p then q”.
• We denote the compound proposition “if p then q”
by p → q.
• The conditional “if p then q” or “p→q” is made up of
two parts;
1. Hypothesis – proposition p in the above conditional.
2. Conclusion – proposition q in the above conditional.
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Example Conditional
1. Given p: John is not at work and q: John is
sick. Find p→q.
p→q: If John is not at work then John is
sick.
2. Given the conditional, If a number is
divisible by four then it is even. Identify the
hypothesis and the conclusion.
– Hypothesis: A number is divisible by four.
– Conclusion: A number is even.
If-Then Statements
• A conditional statement is a statement that can
be written in if-then form.
Example: If an animal has hair, then it is a mammal.
• Conditional statements are always written “if p,
then q.” The phrase which follows the “if” (p) is
called the hypothesis, and the phrase after the
“then” (q) is the conclusion.
• We write p → q, which is read “if p, then q”.
Example 1:
Identify the hypothesis and conclusion of the
following statement.
If a polygon has 6 sides, then it is a hexagon.
If a polygon has 6 sides, then it is a hexagon.
hypothesis
conclusion
Answer: Hypothesis: a polygon has 6 sides
Conclusion: it is a hexagon
Your Turn:
Identify the hypothesis and conclusion of each
statement.
a. If you are a baby, then you will cry.
Answer: Hypothesis: you are a baby
Conclusion: you will cry
b. To find the distance between two points, you can use
the Distance Formula.
Answer: Hypothesis: you want to find the distance
between two points
Conclusion: you can use the Distance Formula
If-Then Statements
• Often, conditionals are not written in “if-then”
form but in standard form. By identifying the
hypothesis and conclusion of a statement, we
can translate the statement to “if-then” form
for a better understanding.
• When writing a statement in “if-then” form,
identify the requirement (condition) to find
your hypothesis and the result as your
conclusion.
Example 2:
Identify the hypothesis and conclusion of the
following statement. Then write the statement
in the if-then form.
Distance is positive.
Sometimes you must add information to a statement.
Here you know that distance is measured or determined.
Answer: Hypothesis: a distance is determined
Conclusion: it is positive
If a distance is determined, then it is positive.
Your Turn:
Identify the hypothesis and conclusion of each
statement. Then write each statement in if-then form.
a. A polygon with 8 sides is an octagon.
Answer: Hypothesis: a polygon has 8 sides
Conclusion: it is an octagon
If a polygon has 8 sides, then it is an octagon.
b. An angle that measures 45º is an acute angle.
Answer: Hypothesis: an angle measures 45º
Conclusion: it is an acute angle
If an angle measures 45º, then it is an acute
angle.
Converse, Inverse, and
Contrapositive
• From a conditional we can also create
additional statements referred to as related
conditionals. These include the converse, the
inverse, and the contrapositive.
Converse, Inverse, and
Contrapositive
• Writing the Converse, Inverse &
Contrapositive of an Conditional
• Given the conditional p→q
– Converse is q→p (reverse the hypothesis and the
conclusion).
– Inverse is ~p→~q (negate both the hypothesis
and the conclusion).
– Contrapositive is ~q→~p (reverse and negate
the hypothesis and the conclusion).
Converse, Inverse, and Contrapositive
Example
• p: a quadrilateral is a rectangle
q: a quadrilateral is a parallelogram
p→q: if a quadrilateral is a rectangle then a
quadrilateral is a parallelogram
Find the converse, inverse and contrapositive
in both logic notation and in words.
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Solution
• Converse; q→p: if a quadrilateral is a
parallelogram then a quadrilateral is a
rectangle.
• Inverse; ~p→~q: if a quadrilateral is not a
rectangle the a quadrilateral is not a
parallelogram.
• Contrapositive; ~q→~p: if a quadrilateral is
not a parallelogram then a quadrilateral is not
a rectangle.
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Truth Value of Contrapositive
• Law of Contrapositives: If an conditional is
true, then it’s contraposive is also true and
conversely (ie. The contrapositive always has
the same truth value as the original
conditional).
Converse, Inverse, and Contrapositive
• Statements that have the same truth value are said to be
logically equivalent. We can create a truth table to
compare the related conditionals and their relationships.
Conditional
Converse
Inverse
Contrapositive
p
q
p→q
q→p
~p →~q
~q →~p
T
T
T
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
F
F
T
T
T
T
Example 4:
Write the converse, inverse, and contrapositive of the
statement All squares are rectangles. Determine
whether each statement is true or false. If a statement
is false, give a counterexample.
First, write the conditional in if-then form.
Conditional: If a shape is a square, then it is a rectangle.
The conditional statement is true.
Write the converse by switching the hypothesis and
conclusion of the conditional.
Converse: If a shape is a rectangle, then it is a square.
The converse is false. A rectangle with 𝓁 = 2
and w = 4 is not a square.
Example 4:
Inverse:
If a shape is not a square, then it is not a
rectangle. The inverse is false. A 4-sided
polygon with side lengths 2, 2, 4, and 4 is
not a square, but it is a rectangle.
The contrapositive is the negation of the hypothesis and
conclusion of the converse.
Contrapositive: If a shape is not a rectangle, then it is
not a square. The contrapositive is true.
Your Turn:
Write the converse, inverse, and contrapositive of the
statement The sum of the measures of two
complementary angles is 90. Determine whether each
statement is true or false. If a statement is false, give
a counterexample.
Answer: Conditional: If two angles are complementary,
then the sum of their measures is 90; true.
Converse: If the sum of the measures of two
angles is 90, then they are complementary; true.
Inverse: If two angles are not complementary,
then the sum of their measures is not 90; true.
Contrapositive: If the sum of the measures of two
angles is not 90, then they are not
complementary; true.
Kites
What makes a quadrilateral a kite?
Definition: Kite
A kite is a quadrilateral
that has two pairs of
consecutive
congruent sides, but
opposite sides are
not congruent or
parallel.
Angles of a Kite
You can construct a kite by joining two different
isosceles triangles with a common base and
then by removing that common base.
Two isosceles triangles can form one kite.
Angles of a Kite
• Just as in an isosceles
triangle, the angles
between each pair of
congruent sides are
vertex angles. The
other pair of angles
are nonvertex angles.
Kite Theorem 6.25
• If a quadrilateral is a kite, then its diagonals
are perpendicular.
Kite Theorem 6.26
• If a quadrilateral is a kite, then exactly one
pair of opposite angles is congruent.
Example 7A
A. If WXYZ is a kite, find mXYZ.
Example 7A
Since a kite only has one pair of congruent angles, which
are between the two non-congruent sides,
WXY  WZY. So, WZY = 121.
mW + mX + mY + mZ = 360
73 + 121 + mY + 121 = 360
mY = 45
Answer:
mY = 45
Polygon
Interior Angles
Sum Theorem
Substitution
Simplify.
Example 7B
B. If MNPQ is a kite, find NP.
Example 7B
Since the diagonals of a kite are perpendicular, they divide
MNPQ into four right triangles. Use the Pythagorean
Theorem to find MN, the length of the hypotenuse of right
ΔMNR.
NR2 + MR2 = MN2
(6)2 + (8)2 = MN2
36 + 64 = MN2
100 = MN2
10 = MN
Pythagorean Theorem
Substitution
Simplify.
Add.
Take the square root of
each side.
Example 7B
Since MN  NP, MN = NP. By substitution, NP = 10.
Answer:
NP = 10
Your Turn:
A. If BCDE is a kite, find mCDE.
A. 28°
B. 36°
C. 42°
D. 55°
Your Turn:
B. If JKLM is a kite, find KL.
A. 5
B. 6
C. 7
D. 8
6.25
6.26