Transcript types of reasoning

WHY DO WE
NEED TO
STUDY
GEOMETRY?
WE HAVE TO STUDY
GEOMETRY TO:
TO PREPARE US FOR
HIGHER MATHEMATICS.
WE HAVE TO STUDY
GEOMETRY TO:
UNDERSTAND AND APPRECIATE
OUR NATURAL AND MAN-MADE
ENVIRONMENT.
WE HAVE TO STUDY
GEOMETRY TO:
PROVIDE US WITH MANY
IMPORTANT FACTS OF
PRACTICAL VALUE.
WE HAVE TO STUDY
GEOMETRY TO:
ENHANCE OUR
ANALYTICAL SKILLS
TO ENABLE US TO
EXPRESS OUR
THOUGHTS
ACCURATELY AND
TRAIN US TO
REASON LOGICALLY.
Before one can start to understand
logic, and thereby begin to prove
geometric theorems, one must first
know a few vocabulary words and
symbols.
If two angles have equal measures,
then they are congruent.
If two segments are congruent,
then they have equal measures.
1.
All right angles are
congruent.
If all angles are right, then they
are congruent.
FOR MORE EXAMPLES:
SEE PAGE 59 ,GEOMETRY TEXTBOOK
General Form
If p, then q
p implies q
p only if q
q if p
Example
If x² = 4, then x = 2
x² = 4 implies x = 2
x² = 4only if x = 2
x = 2 if x² = 4
If two angles have equal measures,
then they are congruent.
If two angles are congruent, then
they have equal measures.
FOR MORE EXAMPLES:
SEE PAGE 59 ,GEOMETRY TEXTBOOK
1. "If someone is a woman, then
they are a human"
"If someone is a human, then
they are a woman."
The converse is FALSE because a man is also a
human.
1. if r is "¬". "All men have hair,"
"All men do not have hair“
"No men have hair."
2. Sam is sleeping in class.
“ It is not true that Sam is sleeping
in class”.
“Sam is not sleeping in class."
p ⇒ q read as “If p, then q”.
-p ⇒ -q read as “If NOT p, then
NOT q”.
Like a converse, an inverse does not necessarily have the
same truth value as the original conditional.
The statement is always true with the contrapositive,
but a statement is not logically equivalent to its
converse or to its inverse.
If it is raining then the ground is getting
wet.
If it is not raining then the ground is not
getting wet.
If the ground is not getting wet
then it is not raining.
If the cat will run then the dog will chase the cat.
If the cat will NOT run then the dog will NOT
chase the cat.
If the dog will chase the cat then the cat will run.
If the dog will NOT chase the cat then
the cat will NOT run.
“If p then q”
“If negation of p then negation of q”
“If q then p”
“If negation of q then negation of p”
NOTE: The conditional statement and its
contra positive are logically equivalent
"If and only if p, then q" means
both that p implies q and that q
implies p.
1. p
q
Mom plays the guitar.
Dad plays the piano.
p∧q
" Mom plays the guitar and Dad
plays the piano ."
1. p
q
Mom plays the guitar.
Dad plays the piano.
pVq
" Mom plays the guitar or Dad
plays the piano ."
1.
It is a square or it is a
trapezoid.
It is not a square.
It is a trapezoid.
Get one-half
crosswise
Given the conditional statement, state the
inverse, converse and its contra positive.
“If today is Tuesday then tomorrow is Wednesday”
Suppose p stands for “Hawks swoop” and q
stands for “Gulls glide”.
Express the following in symbolic form each of
the following statements.
1. Hawks swoop or gulls glide.
2. Gulls do not glide.
3. Hawks do not swoop or gulls do not
glide.
4. Hawks do not swoop and gulls do
not glide.
Let’s see if
your
answers are
correct.
“If today is Tuesday then tomorrow is Wednesday”
“If today is not Tuesday then tomorrow is not
Wednesday”
“If tomorrow is Wednesday then today is Tuesday ”
“If tomorrow is not Wednesday then today is
not Tuesday ”
Suppose p stands for “Hawks swoop” and q
stands for “Gulls glide”.
Express the following in symbolic form each of
the following statements.
1. Hawks swoop or gulls glide.
2. Gulls do not glide.
pVq
q
3. Hawks do not swoop or gulls do not
 p V q
glide.
4. Hawks do not swoop and gulls do
pɅq
not glide.
Perfect score
is 7.
Pass your
paper.
Inductive Versus Deductive Reasoning