Inductive and Deductive
reasoning
Mathematical Proofs
Why Proofs?
Mathematicians need proofs to determine with
absolute certainty what is possible. Also, if we don't
insist on proofs, mistakes can creep in that aren't
easily spotted otherwise.
Euclid stated that the sum of the internal angles of all
triangles is 180 degrees — he thought this was so
obvious, we should just accept it.
Mathematicians were struggling with this proof for
hundreds of years. During the 19th century it even
became a bit of an obsession, but everyone failed to
prove that the angles in a triangle always sum to 180
degrees.
The reason is that it isn't always true. It only works if
you draw your triangle on a flat plane. If you draw it
on a sphere, say an orange, the interior angles add up
to more than 180 degrees.
Inductive Reasoning
Inductive Reasoning is the process of drawing a general
conclusion by observing a pattern. This conclusion is called
a conjecture - basically a pretty decent guess.
Validity and Counterexamples of Conjectures
Make a conjecture
How could you validate
your conjecture?
How could a conjecture
Be shown to be invalid?
Make a conjecture
How could you validate
your conjecture?
Conjecture: An 8.5 x 11 piece of paper can be cut so that a
person will fit through it.
Could you validate this conjecture?
Tutorial
Deductive Reasoning
Deductive reasoning is how we can prove conjectures; by
connecting ideas that we know to be true.
Ex:All numbers ending in 0 or 5 are divisible by 5. The number 35 ends
with a 5, so it must be divisible by 5.
Cacti are plants, and all plants perform photosynthesis. Therefore, cacti
perform photosynthesis.
Incorrect example:All farmers like burgers. Jethro likes chicken wings.
Therefore, Jethro is not a farmer.
Use deductive reasoning to reach a conclusion for each
statement:
A person must be 12 years old or over to have a fishing
license. What can be deduced with certainty about each
person?
a) Sally has a fishing license
b) Bill went fishing
c) Lora is 15 years old
d) George is under 12 years old
e) Tim does not fish
Use deductive reasoning to reach a conclusion for each
statement:
All members of the volleyball team are over 6 feet tall.
What, if anything, can you deduce with a certainty about each
person?
a) Sue is on the Volleyball Team
b) Tom is over 6ft tall
c) Mary is 5′6" tall
d) Bert is not on the Volleyball Team
When proving conjectures deductively we use general cases in
our final proof. This is to demonstrate that the proof
applies to all cases.
The difference between consecutive perfect squares is always
an odd number
Statement and Evidence
Choose a number, triple the number, add 6, subtract the
original number, divide by 2 and subtract 3. The result is
the original number.
Left side
Right side
Q.E.D.
"Which was to be demonstrated."
Terminology
Invalid proof: A proof that contains an error in reasoning or
that contains invalid assumptions (fallacy)
Premise: A statement assumed to be true
Circular reasoning: An argument that is incorrect because it
makes use of what you’re trying to prove in the first place.
Proofs that aren’t valid
Conjecture: 4 = 3
Suppose:
a+b=c
This can be rewritten as:
4a – 3a + 4b – 3b = 4c – 3c
If we reorganize:
Factor using distributive property:
Divide both sides by (a + b – c):
Where is the error in deduction?
4a + 4b – 4c = 3a + 3b – 3c
4 (a + b – c) = 3 (a + b – c)
4=3
Reasoning to Solve Problems
Thinking logically can help in problem solving.
Inductive thinking: Simplifying a problem, looking for
patterns, drawing conclusions from observations
Deductive thinking: Using facts
or assumptions to make an
argument, and drawing logical
conclusions from the argument
Suppose you were lost in the woods and came upon a cabin. In
the cabin there was a lantern, candle, wood stove (with
wood) and a match. In what order to you light things? Which
kind of reasoning did you use?
Making game predictions
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