Basic Mathematics for College Students

Use deductive reasoning to solve problems.

As opposed to inductive reasoning, deductive reasoning moves from the general case
to the specific. For example, if we know that the sum of the angles in any triangle is
180°, we know that the sum of the angles of ^ABCshown in the right margin is 180°.
Whenever we apply a general principle to a particular instance, we are using deductive
reasoning.
A deductive reasoning system is built on four elements:

1. Undefined terms:terms that we accept without giving them formal meaning


2. Defined terms:terms that we define in a formal way


3. Axiomsor postulates:statements that we accept without proof


4. Theorems:statements that we can prove with formal reasoning

Many problems can be solved by deductive reasoning. For example, suppose a
student knows that his college offers algebra classes in the morning, afternoon, and
evening and that Professors Anderson, Medrano, and Ling are the only algebra
instructors at the school. Furthermore, suppose that the student plans to enroll in a
morning algebra class. After some investigating, he finds out that Professor Anderson
teaches only in the afternoon and Professor Ling teaches only in the evening. Without
knowing anything about Professor Medrano, he can conclude that she will be his
algebra teacher, since she is the only remaining possibility.
The following examples show how to use deductive reasoning to solve problems.

2

Appendix III Inductive and Deductive Reasoning A-25

EXAMPLE (^5) Find the next shape in the sequence below.
StrategyTo find the next shape in the sequence, we must consider two changing
patterns at the same time.
WHYThe shapes are changing and the number of dots within them are changing.
Solution
The first figure has three sides and one dot, the second figure has
four sides and two dots, and the third figure has five sides and three
dots. Thus, we would expect the next figure to have six sides and four
dots, as shown to the right.
,, ,

...

Self Check 5

Find the next shape in the

sequence below.

Now TryProblem 27

,,,

...

C

B

A

EXAMPLE (^6) Scheduling Classes An online college offers only one
calculus course, one algebra course, one statistics course, and one trigonometry
course. Each course is to be taught by a different professor. The four professors who
will teach these courses have the following course preferences:

1. Professors A and B don’t want to teach calculus.


2. Professor C wants to teach statistics.


3. Professor B wants to teach algebra.

Who will teach trigonometry?

StrategyWe will construct a table showing all the possible teaching assignments.
Then we will cross off those classes that the professors do not want to teach.

Now TryProblem 31