Readings for Session 3 – (Continued)
Definition of a Set
Set:
A set is a well-defined
collection of objects or ideas.
What does it mean for a set to be
well-defined?
A set
is well-defined if there is no ambiguity as to whether or not an
object belongs to it, i.e.,
a set is defined so that
we can always tell what is and what is not a member of the set.
Example:
C = {red, blue,
yellow, green, purple} is well-defined since it is clear what is
in the set.
Example:
In the opening problem on the previous page of this
session, the solution set to the question “Who is the President?” is
not a well-defined collection.
Does it mean only presidents of nations, or does it
include presidents of companies?
Of universities?
Of clubs? Do
we include only those serving now or all who served in the past?
To be “well-defined” the collection description would
have to settle all such questions.
Example: “The collection
of good students at MSUM” is
not a set. What does it mean to be “good?” Does the collection refer
to past or present students? Also, does MSUM refer to Mankato,
Moorhead, or some other school, college, or university? All
these questions indicate the statement is ambiguous, i.e., it is
not clear which students are members of this collection, hence,
the collection is not well-defined.
Example:
“The collection of the currently enrolled students at
Minnesota State University Moorhead with a grade point average
above 3.0”, is a well-defined set since it is clear which
students would belong to the collection.
Example:
“The collection of young people that are residents of
Minnesota” is not a
set. What does it mean be "young"?
To someone that is 80 years old, a person that is 40 is young,
but a 20 year old person may not consider 40 to be young.
The word “young” is an ambiguous term. Hence, the collection is
not well-defined.
Example:
“The collection of residents of Minnesota between the
ages of 18 and 25 years old inclusive” is a well-defined set
since it is fairly clear who would belong to this collection.
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