Download Deductive Reasoning

Proving Conjectures: Deductive Reasoning
Date:
Recall:
We can use Inductive Reasoning to test a Conjecture. After some evidence, we can keep or revise
the conjecture.
Terminology:
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Deductive Reasoning -
based on
Deductive Reasoning uses a proof. A proof is a logical argument showing that a statement is true for
all cases and that no counterexample exists.
Proofs could be in the form of a logical sequence of justifiable statements, a two-column proof,
Venn diagram, etc.
Example:
Compare how inductive reasoning and deductive reasoning can validate the following
conjecture.
The product of a multiple of 3 and a multiple of 4 is a multiple of 12.
Inductive Reasoning
(3)(4) = 12  multiple of 12
(3)(8) = 24  multiple of 12
(6)(4) = 24  multiple of 12 ….etc.
Yesterday we could _______________________________________________________
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Thus, Inductive Reasoning cannot prove the conjecture must be true.
Deductive Reasoning
Natural numbers are 1, 2, 3, 4, 5, …
The product of any two natural numbers, mn, is also a natural number.
A multiple of 3 can be written as 3m, 3 times a natural number.
A multiple of 4 can be written as 4m, 4 times a natural number.
The product of a multiple of 3 and a multiple of 4 is (3m)(4n) = 12mn.
12mn is 12 times a natural number.
The product of a multiple of 3 and a multiple of 4 is a multiple of 12.
Deductive Reasoning has __________________________________________________
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A generalization is a principle , statement, or idea that has general application.
The previous conjecture could be generalized to:
The product of multiples of tow numbers is a multiple of the product of the two numbers.
Example:
Prove the following conjecture is true.
Any three-digit number is divisible by 3 if the sum of its digits is divisible by 3.
A Venn Diagram consists of circles that represent sets and shows every logical relationship between
these sets.
Ex. 1 Use deductive reasoning to make a conclusion from these statements:
“All koalas are marsupials. All marsupials are mammals.”
“All mammals are warm-blooded. Keith is a koala.”
Steps: 1. From the first sentence, do koalas belong to marsupials or vice versa? Group these
situations in circles. Do this for each statement.
2. Make a conclusion based on the groupings.
Ex. 2 All birds have backbones. Birds are the only animals that have feathers.
Rosie is not a bird. What can be deduced about Rosie?
1. Rosie has a backbone.
2. Rosie does not have feathers.
a.
b.
c.
d.
Ex. 3
Neither Choice 1 nor Choice 2
Choice 1 only
Choice 1 and Choice 2
Choice 2 only
All ostriches are birds. All birds have backbones. Birds are the only animals that have feathers.
Floradora is an ostrich. What can be deduced about Floradora?
1. Floradora has a backbone.
2. Floradora has feathers.
a.
b.
c.
d.
Neither Choice 1 nor Choice 2
Choice 1 and Choice 2
Choice 2 only
Choice 1 only
Ex. 5 Try the following number trick with different numbers. Make a conjecture about the trick.
 Start with the last two digits of your birth year.
 Multiply this number by 3.
 Multiply by 7.
 Multiply by 37.
 Multiply by 13.
NOTE: we can represent types of numbers using algebra
EVEN Integers 
where n = 0, 1, 2, …
Try a few:
ODD Integers 
where n = 0, 1, 2, …
Try a few:
Ex. 6 Prove that the sum of two even integers is always even.
Ex. 7 P. 32: Question #7.
HW: p. 31: #2, 5, 8, 9, 10