1. Basic concepts of functions
Function: Let X and Y be sets, and f is the relationship from X to Y. If for any x∈X, there is a unique y∈Y such that <x,y>∈f, then f is from X to Y The function (transformation, mapping) of, denoted as f:X->Y, or
independent variable and function value (image source and image): f:X->Y, if <x,y>∈f, then called x It is the independent variable (image source), and y is the function value (image) of x.
Several relations on A={1,2,3}, which are functions from A to A.
The characteristics of the function diagram:
each node has and only one arc (including the ring) outward.
The characteristics of the functional relation matrix:
each row has and only one 1
Domain, range and co-domain (common domain): f:X->Y
Domain of f (domain): denoted as dom f or Df
range of f (range): denoted as ran f or f(X)
The codomain of f, which Y calls the codomain of f.
Here are a few familiar relationships on the set of real numbers. Which are the functions from R to R?
1.x cannot be equal to 0, so it is not
2.x=1, y has two corresponding values. No
3. Yes
4. No, x<=0 does not correspond to y
5. No, x<0 does not correspond to y
When judging whether it is a function, it must be noted that all x values in the domain have a y value corresponding to it, and the y value is unique.
Two functions are equal.
There are two functions f: A->B, g: C->D, f=g if and only if A=C, B=D, and for any x∈A, there is f(x) =g(x)
means that their domains, codomains, and mappings are the same.
2. Special functions
Constant value function: function f:X->Y, if y0∈Y exists, so that for any x∈X, f(x)=y0, that is,
ranf={y0}, then f is a constant value function
2. Identity function: the identity relationship Ix is a function from X to X, that is, Ix:X->X, which is called the identity function.
Obviously for any x∈X, Ix(x)=x
3. The mapping type of the function
Surjective : f: X-> Y is a function if, for any y ∈ Y, x ∈ X are present, such that f (x) = y, f is called a full shot. That is , the range of the surjective function is Rf=Y.
In the map: f:X->Y is a function, if Rf⊂Y, then f is
the relational matrix of the surjective function in the map : each row has and only one 1 and each column has at least one 1. (This The number of independent variables must be greater than or equal to the number of dependent variables)
3. Incident: f:X->Y is a function, for any x1, x2∈X, if x1≠fx2, f(x1)≠f(x2), or if f(x1)=f(x2) , Then x1=x2 means that
f is incident (injective, one-to-one)
(At this time, the number of independent variables must be less than or equal to the number of dependent variables, otherwise there will be many-to-one)
4. Bijective: f:X->Y is a function. If f is both surjective and incident, then f is bijective, or f is one-to-one correspondence.
Determine the type of the following function.
Theorem: Let X and Y be a finite set. If the number of elements in X and Y are equal, then f:X->Y is incident if and only if it is surjective.
When X and Y are finite sets, as long as f is incident or surject, then f is bijective.
Function compound operation
Since the function is a special relationship, the compound operation of the function is defined as:
Note: Here, g is written to the left of f, so it is called left compound, which is written in order to take care of mathematical habits.
Definition: The compound of two functions is still a function
. The method
of finding the compound operation of the function is the same as the method of finding the compound operation of the relation. You can directly cross the river and dismantle the bridge, or use the relation graph or relation matrix to find it, but pay attention to writing as the left compound
function compound operation Property:
Theorem 2: The compound operation of the function satisfies the associativity.
Proof: It is similar to the proof of the associativity of the compound of the relation, but it should be noted that the definition of function equality is used to prove.
Inverse operation of function
Definition of relational inverse operation: Let R⊆XXY
look at the following example:
Obviously f-1 is not a function.
It can be seen that if a function is not bijective, its inverse is not a function.
Inverse function definition
2. Properties
