Transcript What Comes Next?

What Comes Next?
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Inductive Reasoning
• Inductive Reasoning – a method that uses a
number of specific examples to arrive at a
conclusion.
• The concluding statement you arrive at using
inductive reasoning is called a conjecture.
• A conjecture is believed to be true, but has not
been proven to be so.
Fermat’s Last Theorem
can only be true for n ≤ 2
This was actually a
conjecture, because no one
was actually sure if it was
true.
Pierre de Fermat
1601-1665
Fermat’s Last Theorem
• It took about 350 years
for someone to prove it.
• After that, it was no
longer a conjecture, it
was a theorem.
• Andrew Wiles became a
very rich man by proving
it in 1993.
Andrew Wiles
Conjectures
• Come up with a conjecture about the sum
of two even numbers.
• Come up with a conjecture about the
product of two consecutive natural
numbers.
• Come up with a conjecture about any
number that is divisible by 5.
A Business Example
Ice Cream
Sales
Can you come up with a conjecture pertaining to Ice Cream Sales and their
relation to Temperature?
Fermat’s Last Theorem
can only be true for n ≤ 2
Obviously this was very
difficult to prove to be true.
What could have been done
to show it wasn’t true?
Pierre de Fermat
1601-1665
Conjectures
• To show that a conjecture is true for all cases,
you must prove it.
• To disprove a conjecture, all you need is one
false example.
• A false example is called a counterexample.
• A counterexample can be a number, a drawing
or a statement.
Counterexample
• Conjecture: All numbers ending in 8 are divisible
by 4.
• Counterexample: 18. This conjecture is not true.
------------------------------------------------------------------• Conjecture: The Washington Nationals have
never lost a game in which they scored at least 9
runs.
• Counterexample: On July 20, 2012 the Nationals
lost to Atlanta 11-10.
Logic
• A statement is a
sentence that is either
true or false.
• The Truth Value of a
statement is either T
(true) or F (false).
• Statements are usually
represented by p or q.
Statements
• p: A square is a rectangle.
– Truth Value: T
• q: An orange is a vegetable.
– Truth Value: F
• What about: Where are you going?
– Is this a statement?
Negation
• Given a statement p, another statement,
called the negation of p, can be created by
inserting the word ‘not’, or removing the
word ‘not’.
• Statement (p): A square is a rectangle.
– Truth Value: T
• Negation (~p): A square is not a rectangle.
– Truth Value: F
Truth Tables
• Truth Tables are a
convenient way for
organizing the truth
values of statements.
Statement
Negation
p
~p
• p: Paris is in France.
T
F
F
T
– If a statement is true, its
negation is false.
• p: Paris is in England.
– If a statement is false, its
negation is true.
Compound Statements
• Two or more statements joined by the word and or the
word or form a compound statement.
• A compound statement using the word and is called a
conjunction.
• A compound statement using the word or is called a
disjunction.
• Compound Statements
– A square is a rectangle and an orange is a vegetable.
– A square is a rectangle or an orange is a vegetable.
Conjunction - Example
• p: A square is a rectangle.
– Truth Value: T
• q: An orange is a vegetable.
– Truth Value: F
• Conjunction: p ^ q
– A square is a rectangle and an orange is a vegetable.
• Truth Value: A conjunction is true if, and only if,
both statements are true.
• Is the conjunction above true?
Disjunction - Example
• p: A square is a rectangle.
– Truth Value: T
• q: An orange is a vegetable.
– Truth Value: F
• Conjunction: p  q
– A square is a rectangle or an orange is a vegetable.
• Truth Value: A disjunction is true unless both
statements are false.
• Is the disjunction above true?
Conjunction and Disjunction
Truth Tables
Conjunction
Disjunction
p
q
p˄q
p
q
pq
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
Conjunction: True if both are true.
Disjunction: False if both are false.
Construct a Truth Table for ~p  q
• Step 1: Add a column
for ~p
p
q
~p
T
T
F
T
F
F
F
T
T
F
F
T
Construct a Truth Table for ~p  q
• Step 1: Add a
column for ~p
• Step 2: Add a
column for ~p  q
• Since it is a
disjunction, it will
only be false when
both are false.
p
q
~p
~p  q
T
T
F
T
T
F
F
F
F
T
T
T
F
F
T
T
What Does This Have to Do With
Geometry?
• Statements have no
shape.
• Statements have nothing
to do with lines.
• Statements have no
angles.
• Then why are they
important?
What is Mathematics?
• You are starting to get a taste
for the fact that there is more
to math than numbers.
• Math is about reasoning, and a
central concept in mathematics
is the idea of proof through
fool-proof reasoning.
• Mathematics just might be the
only place in life … maybe in
the entire universe, where you
can truly prove things.
• Proof leads to truth.
Some mathematician, I believe, has
said that true pleasure lies not in
the discovery of truth, but in the
search for it. – Leo Tolstoy