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# 1.4 More Sets

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 ".)

1.50   Examples.

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 set-like 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.

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