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1.44
Definition (Proposition Form.)
Let
be a set. A
proposition form on
is a rule that assigns to
each element
of
a unique proposition, denoted by
.
1.45
Examples.
Let
Then
is a proposition form on
.
is false, and
is true. Note
that
is neither true nor false. A proposition form is not a proposition.
Let

(1.46) 
Then
is a propostion form, and
is true for all
. Note that
is not a proposition, but if

(1.47) 
then
is a proposition and
is true. Make sure that you see the difference
between the right sides of (
1.46) and (
1.47). The placement of the
quotation marks is crucial. When I define a proposition I will often
enclose it in quotation marks, to prevent ambiguity. Without the
quotation marks, I would not be able to distinguish between the right sides of
(
1.46) and (
1.47). If I see a statement like
without quotation marks, I immediately think this is a statement of the form
and conclude that
.
1.48
Notation.
Let
be a set, and let
be a proposition form on
. Then

(1.49) 
denotes the set of all objects
in
such that
is true. (Read
(
1.49) as ``the set of all
in
such that
".)
Variations on this notation are common. For example,
represents the set of all numbers of the form where
.
1.51
Definition (Union, intersection, difference.)
Let
be a set, let
be the set of all subsets of
, and let
be
elements of
. We define the
intersection of
and
by
we define the
union of
and
by
and we define the difference
by
1.52
Examples.
If
and
, then
1.53
Definition (Ordered pairs and triples.)
Let
be objects (not necessarily all different). The ordered pair
is
a setlike combination of
and
into a single object, in which
is
designated as the
first element and
is designated as the
second
element. The ordered triple
is a similar construction
having
for its
first element,
for its second element and
for its third element. Two ordered
pairs (triples) are equal if and only if they have the same first elements, the same
second elements, (and the same third elements). Thus
1.54
Warning.
Ordered pairs should not be confused with sets.
1.55
Definition (Cartesian product, .)
If
and
are sets, we define the set
by
is called the
Cartesian Product of
and
.
1.56
Example.
If
is the set of real numbers, then
is the set of all ordered
pairs of real numbers. You are familiar with the fact that ordered pairs of real
numbers can be represented as points in the plane, so you can think of
or
as being the points in the plane.
Next: 1.5 Functions
Up: 1. Notation, Undefined Concepts
Previous: 1.3 Equality
Index