SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

FUNCTIONS, MAPPINGS, MAPS, TRANSFORMATIONS, OPERATORS

Def. Function (or mapping, map, transformation, operator). Suppose
that to each element in a set A there is assigned, by some
manner or other, a unique element of a set B. We call such
assignments a *function (or mapping, map, transformation,
operator).* If we let f denote these assignments, we write

which reads “f is a function of A into B”. The set A is called
the domain of f and B is called the co-domain of f. If the
function assigns b ε B to a ε A we say b is the image of a. The
image of a is denoted by f(a), which reads “f of a”. It is called
*the value of f at a *or *the image of a under f *. The element a is
called the* preimage *of b. If P is any subset of A, then f(P)
denotes the set of images of the elements of P; and if Q is any
subset of B, then f ^{-1}(Q) denotes the set of elements of A which
are mapped into Q. We call f(P) the image of A and f ^{-1}(Q) the
inverse image or preimage of Q.

Syn. mapping, map, transformation, operator

In a function from a set A into a set B, several elements of A may all image into the same element in B. In Fig. 1 elements a and b both image into 1. Also, the entire set B may not be covered. See Figure 1.

Range of a function. The range of a function consists of those elements of the co-domain that the function maps into. The co-domain consists of the entire set of elements being mapped into. In Fig. 1 the range consists of elements 1, 2, 3 and 5 whereas the co-domain consists of the entire set B. The range of

is denoted by f(A). Note that f(A) is a subset of B.

The objects of sets A and B can be quite arbitrary. Set A could represent integers, real numbers, complex numbers, vectors, matrices, functions, etc.. Likewise for set B.

Examples.

1] The area of a circle is a function of the radius; the sine of an angle is a function of the angle;
the logarithm of a number is a function of the number. The expression y = 3x^{2} + 7 defines y as a
function of x where it is specified that the domain is (for example) the set of real numbers.

2] The matrix equation y = Ax where A is an mxn matrix and x and y are vectors from two different vector spaces defines a function from one vector space into another. The domain consists of vector space V and the co-domain consists of vector space W with x in V and y in W. Matrix A represents the function which can be viewed as an “operator” that operates on one vector to produce another.

3] The integral

is a function that assigns a real number to a real function f(x) defined on the interval [0,1].

Onto function. A function is said to be “onto” if every element in co-domain B is the image of some element in domain A. Several elements of A may, however, map into the same element of B. See Figure 2.

Syn. surjective function, surjection.

One-to-one function. A function is said to be “one-to-one” if every element of domain A maps into a different element of co-domain B. Different elements image into different elements. No two elements image into the same element. However, the whole set of B may not be covered. See Figure 3

Syn. injective function, injection.

Bijective function. A function that is both one-to-one and onto.

Equal functions. If f and g are functions defined on the same domain D and if f(a) = g(a) for every a D, then the functions f and g are equal and we write f = g.

Identity function. Let A be any set. Let the function f : A → A be defined by the formula f(x) = x, that is let f assign to each element in A the element itself. Then f is called the identity function or the identity transformation on A. It is the function I : A → A which leaves every point of A fixed.

Constant function. A function f of A into B is called a constant function if the same element b B is assigned to every element in A. In other words, f: A → B is a constant function if the range of f consists of only one element. See Figure 4.

Example. Let f : R → R be defined by the formula f(x) = 3. Then f is a constant function since 3 is assigned to every element of the domain R.

Product function. Let f be a function of A into B and let g be a function of B, the co-domain of f, into C. See Figure 5. Let a be an element in A. Then its image f(a) is in B which is the domain of g. Accordingly, we can find the image of f(a) under the mapping g, that is, we can find g(f(a)). Thus we have a rule which assigns to each element a A a corresponding element g(f(a)) C. In other words, we have a function of A into C. This new function is called the product function or composition function of f and g and is denoted by

(g f) or (gf)

More briefly, if f: A→ B and g: B → C then we define a function (g f): A → C by

(g f)(a) g(f(a))

Here ≡ is used to mean equal by definition.

Associativity of products of functions. Let if f: A → B, g: B → C and h: C → D.

Then, as illustrated in Figure 6, we can form the product function gf: A → C and then the function h (gf): A → D.

Similarly, as illustrated in Figure 7, we can form the product function hg: B → D and then the function (hg)f: A → D.

Both (h(gf) and (hg)f are functions of A into D. A basic theorem on functions states that these functions are equal.

Theorem. Let

f: A → B, g: B → C and h: C → D. Then

(hg)f = (hg)f

Thus multiplication of functions obeys the Associative Law for multiplication. As a consequence of this theorem we can write

hgf : A → D

without any parentheses.

Def. Inverse function. The function which exactly undoes the effect of a given function.
Let f be a function of A into B and g be a function of B into A. Then g is an inverse of f if gf =
I where I is the identity function. Thus g undoes the effect of f, leaving the set A unchanged. We
denote the inverse of a function f by f ^{-1}. Thus if function f possesses an inverse f ^{-1}, then f ^{-1}f = I.

Existence of inverse functions. A function f may or may not have an inverse. We have seen that a function may assign the same image in B to several elements of A. See Figure 1 above. A function that does this cannot have an inverse. There is no function which will “undo” that kind of mapping. By definition, functions are single valued. For an inverse function to exist for a given function f the mapping must be one-to-one. In addition a mapping may not cover the entire co-domain. This causes an additional problem.

Let us now consider an important concept, the concept of a *group of transformations.* The term
“transformation” means the same as function. The terms are used interchangeably.

Def. Group of Transformations on a set S . Any set G of one-to-one transformations (i.e. functions) of a set S onto itself which meets the axiomatic conditions for being a group i.e.

1) Closure (if transformations f and g are in G, so is their product fg)

2) The associative law holds i.e. f(gh) = (fg)h

3) Existence of an identity element

4) Existence of inverses i.e. if transformation f is in G, so is its inverse f ^{-1}

Note that group G may be either a finite group or an infinite group. No stipulation on that is made.

Consider the following important set: the set of all possible mappings of a set S of n objects onto itself. It meets all of the axiomatic requirements of a group.

Def. Permutation. A one-to-one mapping of a set of n objects onto itself.

Def. Symmetric group S_{n} on n letters. The group of all permutations on n
objects. It corresponds to the set of all possible mappings of a set S of n objects onto itself.

References.

Lipschutz. Set Theory. Chap. 4

Lipchutz. Linear Algebra. p. 121

James and James. Mathematics Dictionary

Birkhoff, MacLane. A Survey of Modern Algebra. p. 119 - 123

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]