Transcript Sections 1.7 and 1.8

By: Cole Srebro, Dillan Patel, Jerry Langan
 Deductive Structure -a system, of thought in which
conclusions are justified by means of previously
proved or assumed statements.
 4 Elements of Deductive Structure
 Undefined Terms (Points, Rays, Lines, Segments)
 Assumptions known as postulates
 Definitions (Reversible)
 Theorems and other conclusions (Can be proved)
Deductive Structure

Postulate -an unproved assumption
› Ex. Through a point not on a line there is exactly
one parallel to the given line.

Definition -states the meaning of a term or
idea
› Ex. If a point is the midpoint of a segment, then it
divides the segment into two congruent
segments
› If a point divides a segment into two congruent
segments, then it is the midpoint of the segment.
(Reverse)



Conditional Statement- “if p, then q” p and q
are declarative statements. The “if” part of the
sentence is called the hypothesis. The “then”
part of the sentence is called the conclusion. If
“p” the “q” can also be written p=>q (p implies
q).
Converse- the converse of p=>q is q=>p. To
write the converse of a conditional statement
you reverse parts p and q.
Biconditional-if a conditional statement and its
converse are both true, the statement is said to
be biconditional.

Write the converse of each of the following
statements:
If C, then D If D, then C
 If peppers, then spicy
If spicy, then peppers
 All cloudy days are depressing. Therefore, since I
was depressed on Friday, Friday was cloudy.

It could have been cloudy but he could have been
depressed from something else.
PRACTICE PROBLEMS(1.7)
1.
2.
3.
Jim is a barber. Everybody who gets his hair cut by
Jim gets a good haircut. Austin got a good haircut.
What can you deduce about Austin?
All dogs are mammals, and all mammals are
vertebrates. Shaggy is a dog. What can be deduced
about shaggy?
When the sun shines, the grass grows. When the
grass grows, it needs to be cut. The sun shines. What
can you deduce about the grass?
Answers to Practice Problems(1.7)
Nothing. Just because Austin got a good haircut
does not mean that Jim cut his hair. This is always
possible, but nothing can be deduced from the
situation.
2. Shaggy is a mammal and a vertebrate.
3. It needs to be cut.
1.
Statements of Logic
Negation-the negation of any statement
“p” is the statement “not p”. The symbol
for “not p” is ~p. In general, ~ ~p=p,
[not (not p)=p].
 Converse- (if q, then p). Review from
section 1.7
 Inverse- (if ~p, then ~ q).
 Contrapositive- (if ~q, then ~p).

Sample Problems
 Ex. Conditional: If you live in Los Angeles, then you
live in California. (True)
Converse: If you live in California, then you live in
Los Angeles. (False)
Inverse: If you don’t live in Los Angeles, then you don’t
live in California. (False)
Contrapositive: If you don’t live in California, then you
don’t live in Los Angeles. (True)
Geometry Sample Problem
 Conditional: If a ray divides and angle into two congruent
angles, then it bisects the angle. (True)
 Converse: If a ray bisects an angle, then it divides the angle
into two congruent angles. (True)
 Inverse: If a ray doesn’t divide and angle into two
congruent angles, then it doesn’t bisect the angle. (True)
 Contrapositive: If a ray doesn’t bisect an angle, then it
doesn’t divide the angle into two congruent angles. (True)
Statements of Logic
 The sample problems on the previous slides
display this important theorem.
 Theorem 3: If a conditional statement is true,
then the contrapositive of the statement is
also true. (If p, then q<=>if ~q, then ~p.)
STATEMENTS OF LOGIC

Chains of reasoning


1.
2.
3.
4.
Ex. If p => q and q=> r, then p => r. This is called the chain rule,
and a series of conditional statements so connected is known as a
chain of reasoning.
Procedure to form a chain of reasoning and write a concluding
statement:
Make a list of contrapositives to the given conditional statements.
Start the chain with a variable used only once in the given
conditionals.
Continue making the chain using both the list of conditionals and
the list of contrapositives until the chain is complete.
Write a concluding statement.
Ex.

Conditionals
Contrapositives
g=>e
~e=>~g
~t=>w
~w=>t
t=>~e
e=>~t
 Chain: g=>e=>~t=>w or
~w=>t=>~e=>~g
 Concluding statement: g=>w or ~w=>~g

1.
Practice Problems(1.8)
Chain of Reasoning




˜x=>˜w
u=>t
x=>˜t
v=>w
2. Write the converse, the inverse, and the
contrapositive of each statement: If a point is the
midpoint of a segment, then it divides the segment
into two congruent segments.
Chain of Reasoning
˜x=>˜w
w=>x
u=>t
˜t=>˜u
x=>˜t
t=>˜x
v=>w
˜w=>˜v
Chain…u=>t=>˜x=>˜w=>˜v
Conclusion … u=>˜v
1.
Answers to Practice Problems
Continued (1.8)
2.
Converse: If a point divides a segment into two
congruent segments, then it is the midpoint of that
segment.
Inverse: If a point is not the midpoint of a segment,
then it doesn’t divide that segment into two
congruent segments.
Contrapositive: If a point doesn’t divide a segment
into two congruent segments then it is not the
midpoint of that segment.
WORKS CITED
Rhoad, Richard, George Milauskas, and Robert
Whipple. Geometry for Enjoyment and
Challenge. Boston: McDougal Littell & Company,
1991. Print.
 Sparknotes. Sparknotes. Web. 17 January 2010.
