Transcript Geometry as a Deductive System or Structure D Inductive Reasoning

Geometry
as a
Deductive System or Structure
D Inductive Reasoning –
observe specific cases to form a
general rule
Example: Little kid burns his
hand five days in a row and
forms a general rule:“the stove
is always hot and dangerous.”
D Deductive Reasoning –
apply a (known) general rule to
a specific case
Example: High school physics student
applies facts that electricity creates
friction and friction produces heat to
know that he should not touch the
stove when electricity is flowing to the
burner.
*Our two-column proofs use
Deductive Reasoning!
Every specific statement in the
left column is justified by a
known reason in the right
column!
*Our two-column proofs use
Deductive Reasoning!
5 Categories of Reasons in a Proof
Given
Definitions
Properties
Postulates
Theorems (previously proven)
D Conditional Statement or
Implication –
A mathematical statement in
“if – then” format
D Conditional Statement or
Implication A mathematical statement in
“if – then” format
D Hypothesis – the if clause
(becomes the given in a proof)
D Conclusion – the then clause
(becomes the prove statement in a proof)
LOGIC SYMBOLS:
read aloud as
“if…then” or “implies”
read aloud as
“if and only if” or
“is equivalent to”
LOGIC SYMBOLS:
~ read as “not” or
“the negation of”
a, b, c, n, m, …
We use letters as logic variables.
Each represents some “phrase”
Examples of a conditional:
a = today is Thursday
b = tomorrow is Friday
a
b
could be read as:
“If today is Thursday, then
tomorrow is Friday.”
Examples of a conditional:
a = today is Thursday
b = tomorrow is Friday
~a
b
read as “If today is NOT Thursday,
then tomorrow is Friday.”
D Converse – of an implication
swaps the hypothesis and
conclusion
For a b, the CONVERSE is b  a
read as:
“If tomorrow is Friday, then
today is Thursday.”
**Here the original statement is true AND the
converse is also true.
D Inverse – of an implication
negates the hypothesis and
conclusion
For a b, the INVERSE is ~a ~b
read as “If today is not Thursday,
then tomorrow is not Friday.”
**Here the original statement is true AND the
inverse is also true.
D Contrapositive – of an
implication is the converse and
inverse at the same time
For a b, the CONTRAPOSITIVE
is ~ b  ~ a
read as “If tomorrow is not Friday,
then today is not Thursday.”
**Here the original statement is true AND
the contrapositive is also true.
*Converse of definitions are
always true so we say that
definitions are reversible.
*Converse of other true
statements (e.g. theorems) may
be true or false. Anything
goes, so you must prove them!
*Inverse of a true statements (e.g.
theorems) may be true or false.
Anything goes, so you must prove
them!
*Inverse of a true statements (e.g.
theorems) may be true or false.
Anything goes, so you must prove
them!
*Contrapositive of a true statement
is always true.
*Contrapositive of a false statement
is always false.
D Chain of Reasoning –
a sequence of conditional
statements that can be
summarized as one
implication.
ab
bc
cd
de
This is logically equivalent to _?_
* To form a chain of reasoning, look
for the “singletons” to use as the
start and finish of your chain.
•To make the “daisy chain” or
domino effect work, you may swap
a statement with its contrapositive.
(But converses or inverses would be an
illegal swap because they could be
false.)
Form a chain of reasoning and give
a one statement summary:
n  r g  ~r z  ~e ~n  z
This is logically equivalent to _?_
Form a chain of reasoning and give
a one statement summary:
n  r g  ~r z  ~e ~n  z
g  ~r
~r  ~n
~n  z
z  ~e
This is logically equivalent to _?_
Form a chain of reasoning and give
a one statement summary:
n  r g  ~r z  ~e ~n  z
g  ~r
~r  ~n
~n  z
z  ~e
This is logically equivalent to
g ~e
A Venn Diagram can be used to
show set and subset
relationships. Often useful
illustrations in logic.
Ex. If Bob lives in Tosa, then he
lives in WI.
Ex. If Bob lives in Tosa, then he
lives in WI.
People that live in the US
People
that live
in Tosa
People that live in WI
Ex. If Bob lives in Tosa, then he
lives in WI.
People that live in the US
Bob
People
B that live
in Tosa
People that live in WI
Ex. Use the Venn diagram to see which
quadrilaterals are rectangles and rhombuses at the
same time.
Quads
trapezoids
parallelograms
squares
rectangles
rhombuses