文档介绍：Chapter 5 Relations and Functions
Wen-Hsiang Lu (盧文祥)
Department puter Science and Information Engineering,
National Cheng Kung University
2006/03/23
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Chap 5 Relations and Functions
Cartesian Products and Relations
For sets A, B, the Cartesian product (cross product), of A and B is denoted by
Extension of the Cartesian product:
Example : is recognize as the real plane of coordinate geometry and two-dimensional calculus.
The subset is the interior of the first quadrant of this plane.
R3 represents Euclidean three-space, where the three-dimensional interior of any sphere, and two-dimensional planes, and one-dimensional lines are subsets of importance.
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Chap 5 Relations and Functions
Cartesian Products and Relations
Example : Let A = {2, 3, 4}, B = {4, 5}. Then
Definition : For sets A, B, any subset of is called a (binary) relation from A to B. Any subset of is called a (binary) relation on A.
Example : The following are some of the relations from A to B.
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Chap 5 Relations and Functions
Cartesian Products and Relations
Example :
Theorem : For any sets Ų :
Proof
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Chap 5 Relations and Functions
: Functions: Plain and One-to-One
Definition : , a function (mapping) from A to B, is a relation from A to B in which every element of A appears exactly once as the ponent of an ordered pair in the relation.
f(a) = b when (a, b) is an ordered pair in the function f.
(a, b) f, b is called the image of a under f, whereas a is a preimage of b.
f is a method for associating with each a A the unique element f(a) = b B.
(a, b), (a, c) f, implies b = c.
Example :
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Chap 5 Relations and Functions
Functions: Plain and One-to-One
Definition : Function , A is called the domain of f and B the codomain of f .
The subset of B consisting of those elements that appear as ponents in the ordered pairs of f is called the range of f and is also denoted by f(A) because it is the set of images (of the elements of A) under f.
A