ONT Re: Set Theory
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Note 16
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| Elementary Set Theory
|
| Functions
|
| Intuitively, a function is to be identical with the class of ordered pairs
| which is its graph. All functions are single-valued, and consequently two
| distinct ordered pairs belonging to a function must have different first
| coordinates.
|
| 63. Definition.
|
| f is a 'function' if and only if f is a relation
| and for each x, each y, each z, if (x, y) in f
| and (x, z) in f then y = z.
|
| 64. Theorem.
|
| If f is a function and g is a function so is f o g.
|
| 65. Definition.
|
| domain f = {x : for some y, (x, y) in f}.
|
| 66. Definition.
|
| range f = {y : for some x, (x, y) in f}.
|
| 67. Theorem.
|
| domain $U$ = $U$,
|
| range $U$ = $U$.
|
| Proof. If x in $U$, then (x, 0) and (0, x) belong to $U$
| and hence x belongs to domain $U$ and range $U$. þ
|
| JLK, Gen Top, page 260.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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