Matematica delle superiori, Esercizi di Matematica

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MATHEMATICS

Unit 1: Set theory and operations

Unit 2: Functions

Unit 3: Natural and integer numbers

Unit 4: Rational numbers

Unit 5: Real numbers

Set theory

A set is a collection of well defined and distinct objects,

considered as an object in its own right.

Venn Diagram

A = {x,y,z,t..}

A

Symbology

sets: upper case letters

elements: lower case letters

belong to: ∈

not belong to : ∉

x

y

z

t

Subset

B is a subset of A if every element of B is an element of A

Representation

Symbology

A

B

B ⊆ A o A ⊇ B

C ⊄ A

Empty set:∅ ∅ ⊆

A , ∅ ⊆ B, ∅ ⊆ C

Complement (set theory)

The relative complement of A with respect to a set B , is the set of

elements in B but not in A

A

Symbology

Complementary of B : B

B ={x ∈A e x ∉ B}

C

B

Representation

Set operations

Definition symbology representation

Intersection

Union

C = { x : x ∈ A e x ∈B} C = A ∩ B

C = { x : x ∈ A o x ∈B} C = A ∪ B

Unit 2

Functions

Functions

f(x

1

)=f(x

2

)

f(x

3

)

Definition (function): A function , denote it by f , from a set X to a set Y is a relation from X

to Y that satisfies:

• for each element a in X , there is an element b in Y such that < a, b > is in the relation, and
• if < a, b > and < a, c > are in the relation, then b = c****.

The set X in the above definition is called the domain of the function, and Y its codomain.

X

Y

f(X)

x

1

x

2

x

3

x

4

x

5

x

6

f(x

4

)

y

I.E.

X = domain; D.D. = domain of definition f (X) = codomain

Real Function of real variable

f(D.D.) ⊆ R D.D. ⊆ R

y:dependent variable independent variable

Analytic Function : the correspondence x → y is

represented by a mathematic formula that with a fixed

value of x ∈ D.D. allows to univocally calculate the

corresponding value of y ∈ f ( D.D.).

Symbology: y = f ( x )

Some characteristics of the graph of a function

x

f(x)

f(x)

f(x)

f(x)

x

These graphs are not functions Graph of a function

A B C

D

• Each line parallel to the x-axis

meets the graph in only one point

• In A , B , C f(x)=
• The ordinate of point D is f(0).

x

x

x

y

y

y

trascendent

Function classification

functions

algebraic

rational irrational

entire fractional entire fractional

Natural numbers = {0,1,2,3,4…}

representation on the number line

0 1 2 3 4 5 6 7 8 9 10 11 12

ordered

discrete

Allowed operations

The result is a

natural number

Addition

Multiplication

Exponential with a natural exponent

Multiples, submultiples, and divisors

a = b * q

a is multiple of b

a is divisible for b

b is divisor of a

b is a submultiple of a

G. C. D.( a,b )

is the largest positive integer that divides a and

b without a remainder

common factors with the lowest exponent

l.c.m.( a,b )

is the smallest positive integer that is

divisible by both a and b.

Common and not common factors with the highest

exponent

If

or or

Addition

Subtraction

Multiplication

Exponential with a

natural exponent

Exponent

Base

Even Odd

Positive + +

Negative +

Algebric

addition

Properties

a

n

a

m

= a

n+m

a

n

/a

m

= a

n-m

(a

n

m

=a

nm

a

n

b

n

=(ab)

n

a

n

/ b

n

=(a/b)

n

Unit 4

Rational numbers