Download Section 1.4 Proving Conjectures: Deductive Reasoning

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October 10, 2013
Section 1.4
Proving Conjectures:
Deductive Reasoning
May 9­10:15 AM
Proof: A mathematical argument showing that a
statement is valid in all cases, or that no counterexample
exists.
Generalization: A principle, statement or idea that has
general application.
To prove a conjecture is true for all cases, we use deductive
reasoning.
Drawing a specific conclusion
Deductive Reasoning:
through logical reasoning by starting with general assumptions
that are know to be valid.
Sep 12­12:56 PM
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3 Strategies to prove conjectures:
1.
Venn Diagrams (a visual representation)
2.
Number Theory Proofs (choosing a variable to algebraically
represent a situation).
3.
Two-Column Proofs (using statements and reasons in an organized
list). *Will be done in chapter 2*
Sep 12­1:13 PM
1. Venn Diagrams
An illustration that uses overlapping or non-overlapping circles to
show the relationship between groups of things.
Example:
All zips are zaps.
All zaps are zops.
Shaggy is a zip.
What can be deduced about Shaggy?
Since Shaggy is a Zip, we also know he is a zap and a zop.
Sep 12­1:17 PM
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Examples:
1.
All dogs are mammals. All mammals are vertebrates. Shaggy is a
dog. What can be deduced about Shaggy?
2.
Casey voted in the last election. Only people over 18 years old
vote. What can be concluded about Casey?
3.
4.
5.
Mammals have fur (or hair). Lions are classified as mammals.
What can be deduced about lions?
Jun 12­12:25 PM
Sep 12­1:25 PM
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Sep 12­1:21 PM
2. Number Theory
This strategy involves choosing a variable or variables to algebraically
represent a situation.
Even Integers: 2n, where n is an integer.
Odd Integers: 2n + 1, where n is an integer.
Consecutive Integers: n, n + 1, n + 2, etc, where n is an integer.
Consecutive Odd Integers: 2n + 1, 2n + 3, 2n + 5, etc, where n is an
integer.
(By adding two more to the previous number you will get the next
consecutive odd integer.)
Consecutive Even Integers: 2n, 2n + 2, 2n + 4, etc, where n is an
integer.
Sep 12­3:42 PM
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Write an expression for the following:
1) an even number
2) an odd number
3) the sum of two consecutive even integers
4) the sum of the squares of two consecutive integers
5)
The difference of squares of two consecutive integers.
6)
a two digit number
7)
The product of two consecutive integers .
8)
The product of an even and an odd integer.
Sep 17­12:48 PM
Using deductive reasoning to prove your conjecture
Three Steps:
1. Define the variables (let n equal...).
2. Use algebra to prove the conjecture.
3. State what you have proven.
Sep 12­8:02 PM
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Example 1: (Text page 27)
Jon discovered a pattern when adding consecutive integers:
a) 1 + 2 + 3 + 4 + 5 =15
b) (-15) + (-14) + (-13) + (-12) + (-11) = -65
c) (-3) + (-2) + (-1) + 0 + 1 = -5
He claims that whenever you add five consecutive integers, the sum
is always 5 times the median of the numbers.
Prove that Jon‛s Conjecture is true for all integers.
Check Jon‛s examples:
a)
1 + 2 + 3 + 4 + 5 =15
Median # = 3 x 5 = 15
b)
(-15) + (-14) + (-13) + (-12) + (-11) = -65
Median # = -13 x 5 = -65
c)
(-3) + (-2) + (-1) + 0 + 1 = -5
Median # = -1 x 5 = -5
Try a sample with larger numbers:
1233 + 1234 + 1235 + 1236 + 1237 = 6185
1235 x 5 = 6175
Jun 12­12:16 PM
How can you prove that Jon's conjecture is true for all integers?
Prove using generalizations:
Let x = the first number
x + (x+1) + (x+2) + (x+3) + (x+4)
= 5x + 10
= 5 (x+2)
Since the sum equals five times the median number, Jon‛s
conjecture is true.
Sep 12­1:05 PM
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Reflect: Jon used inductive reasoning to develop his conjecture.
We used deductive reasoning to prove Jon‛s conjecture.
Jun 12­12:19 PM
Example:
Conjecture: If you multiply two even integers then the product
will be even.
Let 2m = one even integer
Let 2n = a second even integer
The product = 2m X 2n
= 4mn
= 2(2mn)
By showing the product has a factor of 2 you are proving
that it is even.
Therefore, if you multiply two even integers then the product
will be even.
Sep 12­8:01 PM
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Example:
Conjecture: If you multiply two odd integers then the product
will be odd.
Let 2m + 1 = one odd integer
Let 2n + 1 = a second odd integer
The product = (2m + 1) X (2n + 1)
= 4mn + 2n + 2m + 1
= 2(2mn + n + m) + 1
By showing the product is 2 times an integer plus 1 you are proving
that it is odd.
Therefore, if you multiply two odd integers then the product
will be odd.
Sep 12­8:08 PM
For each example support the conjecture inductively(showing three
examples) and then prove it deductively:
a)
Conjecture: The sum of four consecutive integers is equivalent
to the first and last integers added, then multiplied by two.
Sep 12­8:21 PM
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b) Conjecture: The sum of four consecutive integers is equivalent to
the first and last integers added, then multiplied by two.
Sep 21­10:25 AM
c) Conjecture: The square of the sum of two positive integers is
greater than the sum of the squares of the same two
integers.
Sep 12­8:22 PM
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Example:
Think of any number. Multiply that number by 2, then add 6, and
divide the result by 2. Next subtract the original number.
What is the result?
Sep 12­8:40 PM
Inductive
So we might form a conjecture that the result will
always be the number 3. But this doesn‛t prove the conjecture, as we‛ve
tried only two of infinitely many possibilities.
Deductive
Let x = a number
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Inductive
Deductive
Let x = a number
Sep 12­8:43 PM
Text pg. 31-33
#'s 1-11, 13, 14, 16,17, 19, 20
Mid-Chapter Review: page 35
#'s 2, 5, 8-11
Jun 12­12:29 PM
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