Download Logic

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Integer triangle wikipedia, lookup

History of trigonometry wikipedia, lookup

Line (geometry) wikipedia, lookup

Triangle wikipedia, lookup

Euler angles wikipedia, lookup

Rational trigonometry wikipedia, lookup

Pythagorean theorem wikipedia, lookup

Perceived visual angle wikipedia, lookup

Trigonometric functions wikipedia, lookup

Euclidean geometry wikipedia, lookup

Transcript
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