1.2
REASONING MATHEMATICALLY
1.2.1 Inductive reasoning : uses information from specific examples to draw a general conclusion
Example. On Monday you observe your neighbor walking her dog at
A generalization is the general conclusion
formed from the specific examples.
1.2.1.1 Saturday morning you see your neighbor walking the
dog at
called a counterexample.
1.2.1.2 Studying patterns reveals much vital information in
many areas. In fact, mathematics has been called the science of studying
patterns.
1.2.1.3 Sequences are patterns involving ordered sets of
numbers.
Examples. 1, 3, 5, 7, … might
be labeled the odd numbers and is easily extended. Upon closer examination, we
find that each term
can be found
by adding 2 to the preceding term. When the same number is added to/subtracted from
the preceding term to obtain the next term, this number is called the common
difference. The sequence is called an arithmetic sequence.
2,
5, 8, 11, … is another example of an arithmetic
sequence whose common difference is 3.
3,
9, 27, 81, … is a sequence obtained by multiplying
the preceding term by 3; the fixed number, 3 in this case, is called
the common
ratio and the sequence is called a geometric sequence.
1,
1, 2, 3, 5, 8, …is called the Fibonacci sequence
because it has neither a common difference or common ratio. What
is the
pattern for generating this sequence? The numbers in the sequence can be found
in nature---pinecones, pineapples, flower petals, sea shells, etc.
4,
7, 11, 18, 29, 47, …is a Fibonacci-like sequence.
Why?
1.2.2 Deductive reasoning: drawing conclusions from given true statements using logic rules
1.2.2.1 If – then statements
are called conditionals. The IF – part of the statement is called the hypothesis
and the THEN – part is called the
conclusion.
Example. If today is Tuesday
(hypothesis), then MATH 3315 meets in KH1105 (conclusion).
In most
cases, when looking at logic tables (e.g. p. 25), the hypothesis is p while the conclusion is q.
1.2.2.2Rules of Logic. There are two rules of logic that we will consider and apply to
conditional statements. When working with these rules,
it is
usual to have a conditional statement that is considered to be true as well as
a paired statement of what happens
in the
situation of the conditional. On these two statements then, we can make a
decision in the situation.
If
a yellow flag is waved over the racetrack, then a caution is in force. A yellow
flag is waved over the racetrack.
Decision:
A caution is in force.
“A
yellow flag is waved over the racetrack” affirms the hypothesis and this means the
conclusion is true, which is the decision from the conditional and its paired
statement. Affirming the hypothesis
applies logic rule A.
If
a yellow flag is waved over the racetrack, then a caution is in force. A
caution is not in force now.
Decision:
A yellow flag was not waved over the racetrack.
“A
caution is not in force now” denies the
conclusion and this means the hypothesis is false, which is the decision
from the conditional and its paired statement. Denying the conclusion applies logic
rule B.
Suppose
the paired statement said, “A yellow flag was not waved over the racetrack.” This statement would deny the hypothesis and
make the conclusion false. The statement that falls from this is the inverse
of the original. So, to draw a conclusion from these statements is invalid
reasoning because it assumes the inverse. Likewise, to
affirm the conclusion leads to invalid reasoning because it assumes
the converse.
1.2.2.3 IFF. “iff” is symbolic for if and only if, and this phrase or
symbol signifies a very strong statement because it means that
both the
hypothesis AND conclusion are true.
The
dog is brown with tiger strips if
and only if the dog’s name is Stormy.
True True
The
equivalent
to this statement is two if –
then statements. OR
If
the dog is brown with tiger stripes, then the dog’s name is Stormy.
If
the dog’s name is Stormy, then the dog is brown with tiger stripes.
Notice
that the two statements are an original and its converse.
1.2.2.4 Logical “and”;
Logical “or”. The “and” or
“or” occurs in the hypothesis of the conditional statement.
If
Sam feeds the dog or takes out the garbage, then Sam can ride his bike.
This
statement means that Sam can ride his bike if he completes one, but not both,
of the chores given in the conditional.
If
Sam feeds the dog and takes out the garbage, then Sam can ride his bike.
This
statement means that Sam can ride his bike if he completes both of the chores
given in the conditional.