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REASONING
LOGIC
INDUCTIVE REASONING:
 Use patterns, tests and experiments to
make conjectures

Red Book, 4.2-4.4
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DEDUCTIVE REASONING:
Use facts, theorems, postulates and
definitions, and laws of logic to form a
logical argument.
You are studying bacteria in biology
class. You make a table and record the
amount of bacteria over different period
intervals. Are you using inductive or
deductive reasoning to make a prediction
of the amount of bacteria over a certain
amount of time?
 INDUCTIVE: TESTING

You are proving that a square is a
rectangle. What reasoning are you
using if you use the definitions and
properties to show that a square is a
rectangle.

DEDUCTIVE: Used definitions and
properties

You diet for 3 weeks and lose 3 pounds.
You conclude that you can lose 20 more
pounds in the next 20 weeks.

You use the rise of 4 and the run of 2
between two points on a line and
conclude that the slope is 2.

INDUCTIVE

DEDUCTIVE

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CONJECTURE
COUNTEREXAMPLE
A CONJECTURE IS AN UNPROVEN
STATEMENT THAT IS BASED ON
OBSERVATIONS.
CAN YOU FIND A
COUNTEREXAMPLE?
1. The quotient of two whole numbers is
a whole number.
 2/4 = .5
 2. The square root of a number x is
always less than x
 square root of ¼ = ½ and ½ > ¼

Negation of a statement

The negation of a statement is the
opposite of the original statement.
It is raining
 It is not raining
 It is Monday
 It is Tuesday
A COUNTEREXAMPLE IS A SPECIFIC
CASE FOR WHICH A CONJECTURE IS
FALSE.
CONDITIONAL STATEMENTS

A conditional statement is a logical
statement that is written in if-then form.
The if statement is the hypothesis
 The then statement is the conclusion

Converse
The converse of a conditional statement
will switch the hypothesis and conclusion
 If it is raining, then I will get wet.



If I get wet, then it is raining.
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INVERSE

The inverse of a conditional statement
will negate the hypothesis and
conclusion.

If it is raining, then I will get wet.
If it is not raining, then I will not get wet.
 ( If it is sunny, then I will stay dry.)

EQUIVALENT STATEMENTS
Equivalent statements are when two
statements are both true or both false.
 If two lines intersect to form a right angle,
then they are perpendicular ( true)


If two lines are perpendicular then they
intersect to form a right angle ( true)
Grades improve when you study








1. Write as a conditional statement.
If you study then your grades will improve
2. Write the converse
If your grades improve then you studied.
3. Write the inverse
If you do not study then your grade will not
improve
4. Write the contrapositive
If you do not improve then you did not study.
CONTRAPOSITIVE
The contrapositive of a conditional
statement is combining the converse and
inverse. That means you switch the
hypothesis and conclusion and you
negate both parts.
 If it is raining, then I will get wet.
 If I do not get wet, then it is not raining.

BICONDITIONAL STATEMENT
A biconditional statement is an “if and only if”
statement, usually abbreviated “iff”.
It is raining if and only if I get wet.
When a conditional statement and its converse
are both true you can write the statement as a
true biconditional
Lines are perpendicular iff they form right angles.
VALIDITY
TRUE OR FALSE

IT ONLY TAKES ONE
COUNTEREXAMPLE TO MAKE A
STATEMENT FALSE.
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TRUE OR FALSE
1. TWO LINES INTERSECT IN AT
MOST ONE POINT
 2. IF WE ARE TWINS THEN WE HAVE
THE SAME BIRTHDAY
 3. IF TWO ANGLES ARE
SUPPLEMENTARY THEN THEIR SUM
IS 180 DEGREES.
4. If x2 = 36 then x = 6.

LOGICAL ARGUMENTS

LAW OF DETACHMENT

LAW OF SYLLOGISM (CHAIN RULE)

INDIRECT REASONING
Ex. All rectangles are ||’grams

1. Rewrite the statement as a conditional

2. Identify the hypothesis and conclusion of
the conditional
3. Write the converse of the conditional and
determine the truth value. If false, provide a
counterexample
4. Write the inverse. Determine the truth
value.
5. Write the contrapositive. Determine the
truth value



LAW OF DETACHMENT
If the hypothesis of a true conditional
statement is true, then the conclusion is
also true.
 If two angles have the same measure,
then they are congruent.
 We know m<A = m<B, therefore we
know that they are congruent.

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LAW OF SYLLOGISM
CHAIN RULE
If two conditional statements are true
and Linked together then the hypothesis
of the first and the conclusion of the last
form a conditional that is true.
 If hypothesis of a then conclusion b
If hypothesis of b then conclusion c are
true then.
 If hypothesis a then conclusion c is true.

If I make a good grade then I get to go
out.
If I go out then I get to see my friends


If I make a good grade then I get to see
my friends.
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Slide 22
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e200281816, 11/2/2009
INDIRECT REASONING
You assume the opposite of the
conclusion and try to find a contradiction.
There is no smallest real number.
Assume there is a smallest real number,
then show that you can always subtract
one and get one smaller, therefore there
is no smallest number.
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