the logic of geometry
DESCRIPTION
The Logic of Geometry. Why is Logic Needed in Geometry?. Because making assumptions can be a dangerous thing. Logic Statement. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/1.jpg)
The Logic of Geometry
![Page 2: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/2.jpg)
Why is Logic Needed in Geometry?
•Because making assumptions can be a dangerous thing.
![Page 3: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/3.jpg)
![Page 4: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/4.jpg)
Logic Statement•Logic statements are used in geometry to correctly interpret and understand the definitions of geometric figures in order
to apply these definitions correctly to geometric proofs
and problems.
![Page 5: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/5.jpg)
Conditional StatementsWritten in “if-then” format or p→q
Conditional statements have two parts:Hypothesis and Conclusion
The part between the “if” and “then” is the hypothesis.The part following the “then” contains the conclusion.
Conditional statements can be either true of false.
![Page 6: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/6.jpg)
Example•If an animal is a poodle, then it is
a dog.What is the hypothesis?
an animal is a poodleWhat is the conclusion?
it is a dog
![Page 7: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/7.jpg)
Is this conditional TRUE or FALSE?
TRUE, Therefore we do not need to do anything!!!
![Page 8: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/8.jpg)
Converse StatementsThe order of the hypothesis and conclusion
is switched or flipped: q→p
Conditional (p→q): If an animal is a poodle, then it is a dog.
Converse (q→p): If an animal is a dog, then it is a poodle.
Is this converse TRUE or FALSE?
![Page 9: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/9.jpg)
FALSE
If a statement is false, a counterexample must be provided.
Counterexample – an example (sentence or picture) that proves a statement is false.
Provide a counterexample for: If an animal is a dog, then it is a poodle.
Lab, Golden Retriever, Beagle, …
Is this converse TRUE or FALSE?
![Page 10: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/10.jpg)
Inverse Statements•An inverse of a statement negates the conditional or original statement. Negate
means to make the opposite. ~p→~q
Conditional (p→q): If an animal is a poodle, then it is a dog.
Inverse (~p→~q): If an animal is not a poodle, then it is not a dog.
![Page 11: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/11.jpg)
Is this inverse TRUE or FALSE?
FALSERemember if the statement is false, you must
provide a ______________________.
Provide a counterexample for:If an animal is not a poodle, then it is not a dog.
Lab, Golden Retriever, Beagle, …
![Page 12: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/12.jpg)
Contrapositive Statements•Contrapositive statements switch and negate
the hypothesis and conclusion. It is both a converse and an inverse.
•Conditional (p→q): If an animal is a poodle, then
it is a dog.
•Contrapositive (~q→~p): If an animal is not a dog, then it is not a poodle.
![Page 13: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/13.jpg)
Is this contrapositive TRUE or FALSE?
TRUE, Therefore we do not
need to provide a counterexample!!!
![Page 14: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/14.jpg)
•The conditional and contrapositive have the same truth value. They are either both true or both false.
•The converse and inverse have the same truth value. They are either both true or both false.
WHAT HAPPENS WHEN ALL THE STATEMENTS ARE TRUE?
Equivalent Statements
![Page 15: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/15.jpg)
Biconditional Statements• If both the conditional and converse
statements are true, then they can be written as a single statement using “if and only if” (iff). Denoted as p↔q
•Valid (true) definitions can be written as biconditional statements.
![Page 16: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/16.jpg)
Biconditional StatementsCan we write our conditional statement as a biconditional statement?
If an animal is a poodle, then it is a dog.
NO, both the conditional and converse must be true, but the
converse is false.
![Page 17: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/17.jpg)
ExampleConsider the conditional statement: If two angles are supplementary, then the sum of
the two angles is 180°.IS THIS A TRUE STATEMENT?
WHAT IS THE CONVERSE?• Converse: If the sum of two angles is 180°, then
the two angles are supplementary angles. IS THIS A TRUE STATEMENT?
CAN WE WRITE THE BICONDITIONAL? WHY OR WHY NOT? IF SO, DO IT!!!
• Biconditional: Two angles are supplementary if and only if the sum of the two angles is 180°.
![Page 18: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/18.jpg)
Another ExampleConditional : If x = 3, then .
IS THIS A TRUE STATEMENT?WHAT IS THE CONVERSE?
Converse: If , then x = 3.
IS THIS A TRUE STATEMENT?CAN WE WRITE THE BICONDITIONAL? WHY OR WHY NOT? IF
SO, DO IT!!!
9x2
9x2
![Page 19: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/19.jpg)
You Try!• Conditional (p→q): If three points lie on the same
plane, then the points are coplanar.• Converse (q→p):
• Inverse (~p→~q):
• Contrapositive (~q→~p):
• If possible, Biconditional (p↔q):
![Page 20: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/20.jpg)
Law of Detachment vs. Law of Syllogism
http://www.youtube.com/watch?v=kuyWgDCZR1U
![Page 21: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/21.jpg)
Law of Detachment
•If p→q is true and p is true, then q must be true.
•Example: If an angle is obtuse, then it cannot be acute. ∠A is obtuse. Therefore, ∠ A cannot be acute.
![Page 22: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/22.jpg)
Law of Syllogism•If p→q and q→r are both true, then p→r is true.
•Example: If the electric power is cut, then the refrigerator does not work.If the refrigerator does not work, then the food is spoiled. Therefore, if the electric power is cut, then the food is spoiled.
![Page 23: The Logic of Geometry](https://reader031.vdocuments.net/reader031/viewer/2022033101/5681617c550346895dd10be7/html5/thumbnails/23.jpg)
Law of Detachment vs. Law of SyllogismDraw a conclusion and determine if the examplesbelow use the Law of Detachment or the Law ofSyllogism.
• Mary is shorter than Debbie.Debbie is shorter than Joan.Joan is shorter than Maria.
• If a student wants to go to college, then the student must study hard. Zoe wants to go to Yale. Conclusion: Zoe must study hard.Law of Detachment
Conclusion: Mary is shorter than Maria.Law of Syllogism.