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Functions-Basic Definitions  
  
1677   03:03 مساءً   date: 14-2-2017
Author : Ivo Düntsch and Günther Gediga
Book or Source : Sets, Relations, Functions
Page and Part : 35-38


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Definition 1.1. A function is an ordered triple 〈f,A, B〉 such that

1. A and B are sets, and f ⊆ A × B,

2. For every x ∈ A there is some y ∈ B such that 〈x ,y〉 ∈ f

3. If 〈x ,y〉 ∈ f and 〈x ,z〉 ∈ f, then y = z; in other words, the assignment is unique in the sense that an x ∈ A is assigned at most one element of B.

A is called the domain of f, and B its codomain.

It is customary to write the function 〈f,A, B〉 as f : A → B. Also, if 〈x ,y〉 ∈ f,  then we will usually write y = f(x), and call y the image of x under f.

The set {y ∈ B : There is an x ∈ A such that y = f(x)} is called the range of f.

 

Observe that the range of f is always a subset of the codomain. Observe carefully the distinction between f(x) and f: Whereas f(x) is an element of the codomain,  f is the rule of assignment, conveniently expressed as a subset of A × B.

Suppose that f : A → B and g : C → D are functions. It follows from the definition of a function that they are equal if and only if

1. A = C,

2. B = D,

3. f = g.

If for a function f : A → B it is clear what A and B are, we sometimes call the function simply f, but we must keep in mind that a function is only properly defined if we also give a domain and a codomain!

Usually, after having agreed on a domain and a codomain, f is given by a rule,

e.g. f(x) = x2 , f(t) = sin t,f(n) = n + 1. As in the previous section

with relations, we sometimes use a diagram to describe a function; for example,  the diagram of Figure 1.1 describes the function f : A → B, where A = {1, −1, 0, −2} is the domain of f,B = {1, 0, 2, 4, } is its codomain, and f = {〈1, 1〉,〈−1, 1〉,〈0, 0〉,〈−2, 4〉}.

Figure 1.1: Function arrow diagram

 

Observe that for all x ∈ A, f(x) = x2 . This can also be indicated by writing x → x2 .

 The definition of a function implies that each element of A is the origin of exactly one arrow; it does not imply that at each element of B is the target of an arrow, or that only one arrow from A points to a single element of B. Functions with these properties have special names, and we shall look at them in a later section.

 

Definition 1.2. Let f : A → B be a function.

1. If A = B and f(x) = x for all x ∈ A, the f is called the identity function on A, and it is denoted by idA.

2. If A ⊆ B and f(x) = x for all x ∈ A, then f is called the inclusion function from A to B, or, if no confusion can arise, simply the inclusion. Observe that, if A = B and f is the inclusion, then f is in fact the identity on A.

3. If f(x) = x for some x ∈ A, then x is called a fixed point of f.

4. If f(x) = b for all x ∈ A, then f is called a constant function.

5. If g : C → D is a function such that A ⊆ C, B ⊆ D, and f ⊆ g, then g is

called an extension of f over C, and f is called the restriction of g to A.

While a function may have many different extension, it can only have one restriction to a subset of its domain. The following diagrams shall illustrate these situations:

Consider the assignment h(x) = (sin x)2 , and suppose that dom h = codom h = R. To find h(x) for a given x, we do two things:

1. First find sin x and set y = sin x;

2. Then find y2.

If we look close enough, we find that we have actually used two functions to find h(x):

1. f : R → R, f(x) = sin x,

2. g : R → R, g(y) = y2,

This shows that h(x) = g(f(x)): We have first applied f to x, found that f(x) is an element of dom g, and then applied g to f(x). This brings us to

 

Definition 1.3. Let f : A → B and g : C → D be functions such that ran f ⊆ dom g. Then the function g ◦ f : A → C defined by (g ◦ f)(x) = g(f(x)) is called the (functional) composition of f and g.

the reason for using a different interpretation of composition for functions is historical, and we shall not go into details.

 

Lemma 1.1. If f and g are functions such that ran f ⊆ dom g, then dom(g ◦ f) = dom f, and codom(g ◦ f) = codom g.

Proof. This follows immediately from the definition of composite functions.

One the most useful properties of functional composition is the following:

Lemma .1.2. Let f : A → B, g : B → C, h : C → D be functions. Then,  h ◦ (g ◦ f) = (h ◦ g) ◦ f, i.e. the composition of functions is associative.

Proof. Let p = h ◦ (g ◦ f), and q = (h ◦ g) ◦ f; to show that two functions are equal we use the remarks following to find their domains and co-domains.

Looking first at p, we see that p is the composite of the functions g ◦ f and h, so,  we will have to look at g ◦ f first. Now, dom(g ◦ f) = dom f = A by Lemma

1.1, thus, dom p = dom h ◦ (g ◦ f) = A. Next,

 codom p = codom h ◦ (g ◦ f) = codom h = D,  also by Lemma 1.1.

Looking at q, we see by a similar reasoning that dom q = A and that codom q = codom(h ◦ g). Since codom(h ◦ g) = codom h, we have codom h = D; thus, we find that dom p = dom q, and codom p = codom q.

All that is left to show is that p = q, i.e. that p(x) = q(x) for all x ∈ A. Let x ∈ A;

then,

p(x) = (h ◦ (g ◦ f))(x)

   = h((g ◦ f)(x))

 = h(g(f(x)))

     = (h ◦ g)(f(x))

         = ((h ◦ g) ◦ f)(x)

= q(x).

Thus, (p,A, D) = (q, A, D)

 




الجبر أحد الفروع الرئيسية في الرياضيات، حيث إن التمكن من الرياضيات يعتمد على الفهم السليم للجبر. ويستخدم المهندسون والعلماء الجبر يومياً، وتعول المشاريع التجارية والصناعية على الجبر لحل الكثير من المعضلات التي تتعرض لها. ونظراً لأهمية الجبر في الحياة العصرية فإنه يدرّس في المدارس والجامعات في جميع أنحاء العالم. ويُعجب الكثير من الدارسين للجبر بقدرته وفائدته الكبيرتين، إذ باستخدام الجبر يمكن للمرء أن يحل كثيرًا من المسائل التي يتعذر حلها باستخدام الحساب فقط.وجاء اسمه من كتاب عالم الرياضيات والفلك والرحالة محمد بن موسى الخورازمي.


يعتبر علم المثلثات Trigonometry علماً عربياً ، فرياضيو العرب فضلوا علم المثلثات عن علم الفلك كأنهما علمين متداخلين ، ونظموه تنظيماً فيه لكثير من الدقة ، وقد كان اليونان يستعملون وتر CORDE ضعف القوسي قياس الزوايا ، فاستعاض رياضيو العرب عن الوتر بالجيب SINUS فأنت هذه الاستعاضة إلى تسهيل كثير من الاعمال الرياضية.

تعتبر المعادلات التفاضلية خير وسيلة لوصف معظم المـسائل الهندسـية والرياضـية والعلمية على حد سواء، إذ يتضح ذلك جليا في وصف عمليات انتقال الحرارة، جريان الموائـع، الحركة الموجية، الدوائر الإلكترونية فضلاً عن استخدامها في مسائل الهياكل الإنشائية والوصف الرياضي للتفاعلات الكيميائية.
ففي في الرياضيات, يطلق اسم المعادلات التفاضلية على المعادلات التي تحوي مشتقات و تفاضلات لبعض الدوال الرياضية و تظهر فيها بشكل متغيرات المعادلة . و يكون الهدف من حل هذه المعادلات هو إيجاد هذه الدوال الرياضية التي تحقق مشتقات هذه المعادلات.