Arithmetic Sequence and Series

Arithmetic Sequence and Series

 

Consider the following sequences.

 

a. 1, 6, 11, 16, ... … …

b. 6, 2, -2, -6, … … … …

c. 3, 4.5, 6, 7.5, … … …

 

In the first sequence, each term is increased by 5 than the preceding term. In the second sequence, each term is decreased by 4 than the preceding term. Each term in the third sequence is increased by 1.5 than the preceding term. In each of the above sequences, the difference between a term and its preceding term is equal or constant. Such a sequence is said to be an Arithmetic Sequence or Arithmetic Progression.

Composite Function

Composite Function

 

Let A = {2, 3, 4}, B = {6, 9, 12} and C = {a, b, c}. If f : A → B and g : B → C are defined by f = {(2, 6), (3, 9), (4, 12)} and g = {(6, a), (9, b), (12, c)},


Composite function gof arrow diagram example

f is the function from A to B such that

 

2 A, f(2) = 6 B

3 A, f(3) = 9 B

4 A, f(4) = 12 B

 

Range of f = {6, 9, 12} = domain of g.

Inverse Function

Inverse Function

 

Let A = {2, 3, 4}, B = {a, b, c} and the function f : A → B is defined by f = {(2, a), (3, b), (4, c)}.


Function f from A to B

Here,

 

2 A, f(2) = a B

3 A, f(3) = b B

4 A, f(4) = c B

 

Also domain of f = {2, 3, 4} and range of f = {a, b, c}.

 

f is one-to-one onto function.

Algebraic Functions

Algebraic Function

 

Algebraic Functions are those functions which involve the sum or difference of algebraic terms. The following are some different types of algebraic functions.

 

a. Constant funciton

b. Identity function

c. Linear function

d. Quadratic function

e. Cubic function

Function

Function

 

Consider two sets A and B. Any non-empty subset R of the cartesian product A × B is called a relation from A to B. A special type of relation associates each element of set A with the one and only element of B. This is indeed a refinement of the concept of relation. Such a refinement is known as a function. We may define a function in the following way:


 

A function from a set A to a set B is a relation (or rule) which associates each element of A with a unique element of B.

 

Symbolically, we write

 

f : A → B

 

to mean “f is a function from A to B”. Further, an element y of B associated with an element x of A is denoted by f(x). Equivalently, we write y = f(x) which reads ‘y equals f of x’. Here f(x) is known as the image of f at x or the value of f at x.

 

The letters f, g, h, F, G, H, φ etc. are reserved for denoting functions.

Relation

Relation

 

If A and B are two non-empty sets then any subset of ordered pairs of a cartesian product A × B is called a relation from set A to set B. A relation is denoted by xRy or simply R, if (x, y) R. A relation from set A to A is called a relation on A.

Cartesian Product

Cartesian Product

 

If A and B are two non-empty sets then the set of all possible ordered pairs (x, y) such that the first component x is an element of A and the second component y is an element of B is called the Cartesian Product of set A and B. It is denoted by A × B which is read as “A cross B”.


 

In the set-builder form, we can write

A × B = {(a, b): a A, b B}

Ordered Pairs

Ordered Pairs


An ordered pair is a pair of objects whose components occur in a special (fixed) order. It is written by listing the two components in a specific order, separating them by a comma, and enclosing the pair in parenthesis (small brackets).

Pythagorean theorem

Pythagorean Theorem

 

Pythagoras (who died about 500 BC) was a Greek philosopher. He was a geometer and a great mystic, who with his followers, seems to have been the first to take mathematics seriously as a study in its own right as opposed to being a collection of formulae for practical calculation. The Pythagoreans are credited with the discovery of the well-known “Pythagoras Theorem” on right-angled triangles.