The Logic of Geometry

Why is Logic Needed in Geometry?

•Because making assumptions can be a dangerous thing.

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.

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.

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

Is this conditional TRUE or FALSE?

TRUE, Therefore we do not need to do anything!!!

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?

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?

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.

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, …

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.

Is this contrapositive TRUE or FALSE?

TRUE, Therefore we do not

need to provide a counterexample!!!

•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

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.

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.

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°.

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

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):

Law of Detachment vs. Law of Syllogism

http://www.youtube.com/watch?v=kuyWgDCZR1U

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.

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.

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.