Chapter 1
Mathematical Reasoning
Section 1.3
Inductive and Deductive Reasoning
Many mathematical ideas are developed using a combination of both inductive and
deductive reasoning. This becomes a two-step process.
In the first step inductive reasoning is used as a “discovery” process when examples
are used to see if a pattern can be recognized or identified.
In the second step deductive reasoning verifies the pattern discovered in the first
step is true under certain assumptions.
In each of the following explain if inductive or deductive reasoning is being used.
a. An employee has worn a blue shirt every Friday they have worked at a company.
A person concludes they will wear a blue shirt this Friday to work. inductive
b. An employee at a company must wear a uniform that has a blue shirt. A person
concludes they will wear a blue shirt this Friday to work.
deductive
c. I am eating at a vegetarian restaurant. I conclude all the dishes on the menu are
meatless.
deductive
d. After reading the entire menu I conclude all of the dishes on the menu are not
meatless
inductive
The key is to determine if some specific examples are being used to draw the
conclusion or some general principle.
We said previously that categorical statements can be written using an if_then_
sentence construction or the only if clause. Consider the following examples.
tall people
What is different about
these two statements?
basketball
players
All basketball players are tall
people.
If a person is a basketball player
then they are tall.
A person is tall, if they are a
basketball player.
A person is a basketball player,
only if they are tall.
basketball players
tall people
Logically (from the
diagram) they represent
different situations.
In terms of grammar the
phrase that is the
hypothesis has
interchanged with the
phrase that is the
conclusion.
All tall people are basketball
players.
If a person is tall then they are a
basketball player.
A person is a basketball player, if
they are tall.
A person is tall, only if they are a
basketball player.
These statements are called converses of each other. The converse of the
statement “If phrase A then phrase B” is “If phrase B then phrase A”.
A statement like an if_then_ statement is called an implication statement. There is
only one situation when it is false. The only time it would be considered false is
when the hypothesis is true and the conclusion is false.
Example
If a person was a US president then they were commander-in-chief of US forces.
true, all presidents serve as commander –in-chief
If a person was a US president then they were elected to office.
false, Ford was not elected
If you can find an instance where the hypothesis is true and the conclusion is false
then the entire statement is false. This reasoning comes up in mathematics in the
following way:
Try these,
If 3x+7=19 then x=4.
true, if 3x+7=19 is true, algebra shows x must equal 4
If x2=16 then x=4.
false, if x2=16 is true, the value of x can be -4
There is one other type of logical statement that comes up often in mathematics and
that is the “if and only if” statement. This statement is true when an if_then_
statement and it converse are both true at the same time.
felines
cats
felines
cats
felines
All cats are felines.
If an animal is a cat then it is a
feline.
cats
This means cats and felines
describe the same thing or we
say they are equivalent.
All felines are cats.
If an animal is a feline then it is
a cat.
An animal is a cat if and only if it
is a feline.
An if and only if statement is used to represent ideas that are the same or
sometimes we say the word equivalent in a certain way.