Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic

Advanced Geometry

Section 1.7 and 1.8- Deductive Structure / Statements of Logic

Learner Objective: Students will write the converse, inverse, and  contrapositive of a conditional statement, determine the truth  value of statements and will use the chain rule to make logical  conclusions.

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Learner Objective: Students will write the converse, inverse, and contrapositive of a  conditional statement, determine the truth value of statements and will use the  chain rule to make logical conclusions.


  

Building blocks of Geometry:

Undefined Terms - Terms which we can't officially  define. We describe their properties but don't  formally define them. examples: point, line, plane

Definitions - State the meaning of a term. There is  never a need to prove a definition. (It is true  because we say it is true.) examples: Def. of a right angle, Def. of a   segment bisector...


  

Building blocks of Geometry (cont.):

Postulates - An unproved assumption. These are pretty  obvious statements which we are unable to prove but  which clearly must be true. example: Two points determine a line.

Theorems - A mathematical statement which can be  proved. example: If two angles are both right angles,  then they are congruent.


Definitions, Postulates and Theorems are all conditional  statements. They have some (sufficient) condition that  leads to a (necessary) conclusion.

If two angles are both right angles,

then they are congruent.

Having two right angles is sufficient evidence to conclude that the angles are congruent.

When two angles are both right angles, it is necessary to conclude that they are congruent (they have to be).

What happens if we reverse the condition and conclusion?


Here is the theorem with the condition and conclusion reversed:

If two angles are congruent

then they are both right angles.

Is knowing that two angles are congruent sufficient evidence to conclude that they must be right angles?

If two angles are congruent, is it necessary to conclude that they are right angles?

Is this a true statement?

For a statement to be considered true, it must ALWAYS be  true. NOT ALWAYS TRUE = FALSE


Conditional statements can be written in "If p then q" form where p is the condition or hypothesis of the statement (trigger) and q is the conclusion (result).

"If p then q" can be symbolized

(also read "p implies q")


Reversing the condition and the conclusion results in the Converse of the original statement. Thus the convers is symbolized

If both the conditional and the converse are true, the statement is said to be reversible.

Definitions are always reversible. Postulates and Theorems are sometimes reversible.


The Negation (opposite) of the statement p is "not p" and is symbolized

The Inverse of the conditional statement is formed when we negate both p and q. Thus the inverse is symbolized


The Contrapositive of the conditional statement is formed by both reversing and negating p and q. Thus, the contrapositive can be symbolized


The conditional statement and it's contrapositive have the same "truth value". That is, they are either both true or both false.The inverse and the converse have the same truth value.

Conditional T T F F

Converse T F T F

Inverse T F T F

Contrapositive T T F F


Example

Conditional:

Converse:

Inverse:

Contrapositive:

If two angles are both right angles, then they are congruent

If two angles _______________then ______________________




Example

Conditional:

Converse:

Inverse:

Contrapositive:

If an angle is an acute angle, then its measure is greater than 0 and less than 90.

If an angle _____________________________, then __________________________________.

If an angle _____________________________, then __________________________________.

If an angle _____________________________, then __________________________________.


A series of conditional statements can be connected together using the Chain Rule.

If and then


Example:

and, , ,

At Hilldale High School, there is a rule that any student caught fighting must be given a three day suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are true?

Bill has been given a three day suspension so we know that he  must have been caught fighting.

Bob has never been caught fighting, so we know that he has  never been given a three day suspension.

Bo has never been given a three day suspension, so we know  that he has never been caught fighting.



Bill has been given a three day suspension so we know that he  must have been caught fighting.


If fighting then suspension.

If suspension then fighting

Conditional:

Converse:

It is not necessary for someone to Fight in order to receive a suspension. Bill could have been suspended for something else. The converse is false.

False




If NOT fighting then NO suspension.

Conditional:

Inverse:

Fighting is a sufficient condition to receive a suspension but not a necessary one. Bob could have been suspended for something else. The inverse is false.

False

Bob has never been caught fighting, so we know that he has  never been given a three day suspension.




If NO suspension then NOT fighting.

Conditional:

Contrapositive:

Since suspension is a necessary result of fighting, NOT being suspended is sufficient to conclude that there was NO fighting. The contrapositive is true.

True

Bo has never been given a three day suspension, so we know  that he has never been caught fighting.



















Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic

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