Section 1.3
Logic
31
Version: Fall 2007
1.3
Logic
Two of the most subtle words in the English language are the words “and” and “or.”
One has only three letters, the other two, but it is absolutely amazing how much
confusion these two tiny words can cause.
Our intent in this section is to clear the
mystery surrounding these words and prepare you for the mathematics that depends
upon a thorough understanding of the words “and” and “or.”
Set Notation
We begin with the definition of a
set
.
Definition 1.
A
set
is a collection of objects.
The objects in the set could be anything at all: numbers, letters, first names, cities,
you name it. In this section we will focus on sets of
numbers
, but it is important to
understand that the objects in a set can be whatever you choose them to be.
If the number of objects in a set is finite and small enough, we can describe the set
simply by listing the elements (objects) in the set. This is usually done by enclosing
the list of objects in the set with curly braces. For example, let
A
=
{
1
,
3
,
5
,
7
,
9
,
11
}
.
(2)
Now, when we refer to the set
A
in the narrative, everyone should know we’re talking
about the set of numbers 1, 3, 5, 7, 9, and 11.
It is also possible to describe the set
A
with words. Although there are many ways
to do this, one possible description might be “Let
A
be the set of odd natural numbers
between 1 and 11, inclusive.” This descriptive technique is particularly efficient when
the set you are describing is either infinite or too large to enumerate in a list.
For example, we might say “let
A
be the set of all real numbers that are greater
than 4.” This is much better than trying to list each of the numbers in the set
A
, which
would be futile in this case. Another possibility is to combine the curly brace notation
with a textual description and write something like
A
=
{
real numbers that are greater than 4
}
.
If we’re called upon to read this notation aloud, we would say “
A
is the set of all real
numbers that are greater than 4,” or something similar.
There are a number of more sophisticated methods we can use to describe a set.
One description that we will often employ is called
setbuilder notation
and has the
following appearance.
A
=
{
x
:
some statement describing
x
}
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Chapter 1
Preliminaries
Version: Fall 2007
It is standard to read the notation
{
x
:
}
aloud as follows: “The set of all
x
such that.”
That is, the colon is pronounced “such that.”
Then you would read the
description that follows the colon. For example, the set
A
=
{
x
:
x <
3
}
is read aloud “
A
is the set of all
x
such that
x
is less than 3.” Some people prefer to
use a “bar” instead of a colon and they write
A
=
{
x

some statement describing
x
}
.
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 Set Theory, Complex number, real line

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