ChFuctionsetc离散数学英文实用PPT课件

where a A and b B, then
domain
A is the domain(定义域) of f
f
B is the co-domain(伴域) of f
b is the image of a
a is the pre-image(原像) of b Range(值域) of f: set of all images of
Propositional Statement?
(y B)(x A)( f (x) y)
x y
f
Example: f: Z Z, f(x) = x2 onto? How about Z+Z+ ?
R+ R+ ?
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A
B
This is not allowed
One-to-One And Onto or One-to-One Correspondence (Bijective双射) Functions
Definition of a Function
A function from set A to set B, denoted by f : A B, is the assignment of exactly one element of B to each element of A
Given f : A B, and if f(a) = b (f maps a to b)
A B
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Functions
• A function f: A → B can also be defined as a subset of A×B (Cartesian product). This subset is restricted to be a relation where no two elements of the relation have the same first element
A function f is in one-to-one correspondence if it is both one-to-one and onto.
Number of elements in A and B must be the same when they are finite. Every element in A is uniquely associated with exactly one element in B. Examples: f: RR, f(x) = -x ? g: Z+Z+, g(x) = x2 ?
• Specifically, a function f from A to B contains one, and only one ordered pair (a, b) for every element a∈ A. x[ [x A y[z) f y = z] ]
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Functions from Functions
If f1 and f2 are two functions from A to R (real numbers), then g = f1 + f2 and h = f1 f2 are also functions from A to R defined by:
The propositional statement?
This is not allowed
xy(( f (x) f ( y) x y)
f
(x y f (x) f (y)))
xy( f (x) f ( y) x y)
Example: f: Z Z, f(x) = x2 one-to-one? g: Z Z, g(x) = x3 one-to-one? A
g(x) = (f1+f2)(x) = f1(x) + f2(x) h(x) = (f1f2)(x) = f1(x) f2(x)
Example: f1(x) = x, f2(x) = x2. (R to R) (f1+f2)(x) = x + x2 (f1f2)(x) = x3
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Functions of Sets
elements of A
pre- a image
co-domain
range
b=f(a)
image
Example: f: Z Z, f(x) = x2 domain/co-domain: Z = set of integers range: perfect squares {0, 1, 4, 9,...}
h: Z+Z+, h(x) = x2 one-to-one?
B
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Onto (Surjective满射) functions
A function f from A to B is onto if for every element b in B
there is an element a in A, f(a)=b. Co-domain = Range
is Y • The pre-images of Z are a, c and d • Let S = {c, d}, f(S) = ? • Let T = {a, b, c}, f(T) = ?
f
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One-to-one (Injective单射) Functions
A function f is one-to-one if and only if f(x) = f(y) implies x = y for all x, y in the domain of f
f: A B, and S is a subset of A. Then we can define f: S Image(S) = f(S)
AS
f
f(S)
B
f (S) { f (s) | s S}
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Function Example
• The domain of f is A = {a, b, c, d} • The codomain is B = {X , Y, Z} • The range of f is {Y, Z} • The pre-image of Y is b. The image of b