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Definition.
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.
Examples.
Which of the following are sets?
1.
The collection of good
students
The collection is not a
set since the word “good” is ambiguous and the phrase is not
well-defined.
2.
The collection of
students at MSUM with a grade point average above 3.0.
It is fairly clear which students would belong to this
collection, hence, it is a set since it is well-defined.
3.
The collection of young
people that are residents of Minnesota
The collection is not a
well-defined set since “young” is an ambiguous term.
4.
The collection of
residents of Minnesota between the ages of 18 and 25 years old
It is fairly clear
who would belong to this collection, hence, it is a well-defined set.
We have three ways of describing sets:
1.
by name or verbal description of the elements of a set,
2.
by roster (list) form by listing the elements separated by commas
and using braces to enclose the list, or
3.
by set-builder notation that uses a variable and a rule to
describe the elements of a set.
Let
A represent the collection of
states that border Minnesota.
A
= {North Dakota, South Dakota, Iowa, Wisconsin, Michigan}
A
= {x :
x is a state bordering Minnesota}
This is read as “A is
the set of all x such that x is a state
bordering Minnesota”.
B
= {x :
x is a counting number}