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Logic
1. To write a conditional
2. To identify the hypothesis and
conclusion in a conditional
3. To write the converse, inverse and
contrapositive of a given conditional
4. To state the truth value of each of the
above (draw conclusions)
5. To write a biconditional
Conditional- an if-then statement
Write a conditional with each of the
following:
•
•
•
•
•
•
A right angle has a measure = 90◦.
If an angle is a rt. <, then it = 90◦.
If an < = 90◦, then it is a rt. <.
Christmas is on December 25th.
If it is Christmas, then it is Dec. 25th.
If it is Dec. 25th, then it is Christmas.
Every conditional has a hypothesis and a
conclusion. The hypothesis always follows the if
and the conclusion always follows the then.
Underline the hypothesis once and the
conclusion twice for the previous
statements.
Conditional- an if-then statement
Write a conditional with each of the
following:
•
•
•
•
•
•
A right angle has a measure = 90◦.
If an angle is a rt. <, then it = 90◦.
If an < = 90◦, then it is a rt. <.
Christmas is on December 25th.
If it is Christmas, then it is Dec. 25th.
If it is Dec. 25th, then it is Christmas.
The following is a Venn diagram.
Use it to write a conditional.
At least a 4 year
college degree
Teacher
If you are a teacher,
then you have at
least a 4 year college
degree.
Write a conditional.
dogs
Chow
If you are a chow,
then you are a dog.
Counterexamples-examples for which a
conjecture (statement) is incorrect.
If it is a weekday, then it is Monday.
counterexample– it could be Tuesday
If the animal is a dog, then it is a poodle.
counterexample--- it could be a lab
If a number is prime it is not even.
counterexample---2 is a prime #
Define converse, inverse, and
contrapositive of a given
conditional.
Converse of a conditional ----flips the
hypothesis and conclusion
Inverse of a conditional-----negates both
the hypothesis and conclusion
Contrapositive of a conditional ----flips and
negates the conditional
Logic Symbols
• Conditional
p→q
• Converse
q→p
Flips conditional
• Inverse
~p → ~q
negates conditional
• Contrapositive ~q → ~p
flips and negates conditional
If 2 segments are congruent, then
they are equal in length.
• Write the converse, inverse,& contrapositive
for the above statement.
Converse---- If 2 segments are equal in length,
then they are congruent.
Inverse-----If 2 segments are not congruent,
then they are not equal in length.
Contrapositive---- If 2 segments are not equal
in length, then they are not congruent.
If 2 angles are vertical, then they
are congruent.
• Write the 1.converse 2. inverse
3. contrapositive.
• If 2 angles are congruent, then they are
vertical.
• If 2 angles are not vertical, then they are
not congruent.
• If 2 angles are not congruent, then they
are not vertical.
Write the 1.converse 2. inverse
3. contrapositive of the following
definition
If an angle is a right angle, then the angle
is equal to 90 degrees.
If an angle is equal to 90 degrees, then it is
a right angle.
If an angle is not a right angle, then it is not
equal to 90 degrees.
If an angle is not equal to 90 degrees then it
is not a right angle.
Go back and determine the truth
values of all your problems. Do you
notice anything?
If 2 segments are congruent, then
they are equal in length.
• Write the converse, inverse,& contrapositive
for the above statement.
Converse---- If 2 segments are equal in length,
then they are congruent. Inverse-----If 2
segments are not congruent, then they are
not equal in length.
Contrapositive---- If 2 segments are not equal
in length, then they are not congruent.
Note the above is a definition!!!!
If 2 angles are vertical, then they
are congruent.
• Write the 1.converse 2. inverse
3. contrapositive.
• If 2 angles are congruent, then they are vertical.
• If 2 angles are not vertical, then they are not
congruent.
• If 2 angles are not congruent, then they are not
vertical.
• Note the above is a theorem!!!!
Write the 1.converse 2. inverse
3. contrapositive of the following
definition
If an angle is a right angle, then the angle
is equal to 90 degrees.
If an angle is equal to 90 degrees, then it is
a right angle.
If an angle is not a right angle, then it is not
equal to 90 degrees.
If an angle is not equal to 90 degrees then it
is not a right angle.
Truth Values
• The conditional and the
contrapositive always have the
same truth value.
• The converse and the inverse
always have the same truth value.
Truth Values
• Note the truth values are all true if
your conditional started with a
definition.
• This is not necessarily true for a
theorem.
D
B
Isosceles Triangle
Theorem
A
If 2 sides of a triangle
are congruent, then the
angles opposite those
sides are congruent.
If DB ≅ DA then, <B ≅ < A.
Converse of
D
B
Isosceles Triangle
Theorem
A
If 2 <‘s of a triangle are
congruent, then the
sides opposite those
angles are congruent.
If <B ≅ < A, then BD ≅ DA.
Biconditional- a statement that combines a
true conditional with its true converse in an
if and only if statement.
Conditional- If an < is a rt <, then it = 90◦
converse If an < = 90◦, then it is a right <.
• An angle is a right angle if and only if it is equal
to 90 degrees.
• An angle is equal to 90 degrees iff it is a right
angle.
Write a biconditional.
• If 3 points lie on the same line, then they
are collinear.
• If 3 points are collinear, then they lie on
the same line.
• 3 points are collinear if and only if they lie
on the same line
• 3 points are on the same line if and only if
they are collinear.
Write a biconditional.
• If 2 lines are skew, then they are
noncoplanar.
• If 2 lines are noncoplanar, then they are
skew.
• 2 lines are noncoplanar iff they are skew.
• 2 lines are skew iff they are noncoplanar.
Write a converse, inverse, contrapositive
and biconditional for the following:
If 2n = 8, then 3n = 12.
Converse If 3n = 12, then 2n = 8.
Inverse If 2n ≠ 8, then 3n ≠ 12.
Contrapositive If 3n ≠ 12, then 2n ≠ 8.
2n = 8 iff 3n = 12
3n = 12 iff 2n = 8
•Note every
definition is
biconditional!
Rewrite as 2 if-then statements (conditional
and converse)
(x+4) ( x-5) = 0 iff x= -4 or x= 5
If (x+4) (x-5) = 0 then x= -4 or x= 5.
If x = -4 or x = 5, then (x+4) ( x-5) = 0.
Write the converse of the given
conditional, then write 2 biconditionals
• 1. If a point is a midpoint, then it divides a
segment into 2 congruent halves.
If a point divides a segment into 2 ¤ halves, then
it is a midpoint.
A pt. is a midpt iff it divides a segment into 2 ¤
halves.
A pt. divides a segment into 2 ¤ halves iff it is a
midpoint.
Assignments
Homework---pp.71-73 (2-4;9-12;15-29;3335) p. 78 (1-11 0dd) p 267 (1-9 odd)
Classwork– HM worksheet # 11