"Minus times minus is plus"

“How (-)×(-) = + ”?

As we all know that (plus × plus) is plus and it’s easy to understand but have you ever wondered that how (minus × minus) is plus.

Many of us are using this condition in our daily mathematical life knowingly and unknowingly but you may have never thought that how this product of two negative become a positive. Well, I’m going to show you logically how this “-x-‘’ become “+” in a easy way .There might be many other proofs regarding this awesome thing but I’m going  to explain in a simple ways.

Firstly, you must have the knowledge of simple things like this (+ x-= -) and (-x+ =-)

First, let us start with easiest way, (besides mathematics is all about the learning of patterns).

(-1) x (+3) =-3

(-1) x (+2) =-2

(-1) x (+1) =-1

(-1) x (0) = 0

(-1) x (-1) =+1

(-1) x (-2) =+2

(-1) x (-3) =+3

               From the above pattern it’s easy to understand how negative times negative is positive. It sounds total mathematical? How do you feel?

Here we can go with proof which is kind of language type.

Let’s suppose a town, consider the good guys as positive (+) or bad guys as negative (-) in the town. If the guys enter the town take that as positive (+) and if the guys leave the town take that as negative (-):

Now again we go through the following pattern by considering the above assumption:

•        If the good guys (+) enter (+) the town, that is good (+) for the town.            (+) x (+) =+

•        If the good guys (+) leave (-) the town, that is bad (-) for the town.                   (+) x (-) =-

•        If the bad guys (-) enter (+) the town, that is bad (-) for the town.                      (-) x (+) =-

•        If the bad guys (-) leave (-) the town, that is good (+) for the town.                     (-) x (-) =+

So, these were the basic two proofs!

Here are some other which will help you in different way to understand the problem!

We all know that

                         -1+1=0.

 Multiplying this equation by -1, we get:

(-1)*[(-1)+1]=(-1)*(0).

Using the distributive property, we get,                      

(-1)(-1)+(-1)1=(-1)0                                                  [ a(b+c)=ab+ac (distributive property)]

As we know,

                         (-1)1=-1 and (-1)0=0

So, by applying this and adding 1 on both sides,

                         (-1)(-1)+(-1)+1=0+1

                   Or, (-1)(-1)+0=1

                   Or, (-1)(-1)=1

This proves the relationship.

At last but not the least:

From the index rule.

 We know that

(a^x)^y = a^{x y},

Which holds true for a=2 and x=y= -1.

Therefore,

(2^-1)^{-1 =} (2)^(-1)(-1).

By the definition of negative exponent, we get

(2^-1)^-1 =(1/2)^-1  =2.

We can then conclude that 2¹=2^(-1)(-1)

•       1=(-1)(-1)

As we know if the bases are equal, so must the exponents be equal.

These were the illustration that proves how the product of two negative becomes positive.

If you have got any other ways you can comment down.

Thank you!

Domain Codomain and Range

In this article in short, we will talk about domain, codomain and range of a function. In previous article we have talked about function and its type, you can read this here.

Domain, Codomain and Range:

Domain:

The set of input values for which the function is defined is called Domain of a function. That is the function provides output for every elements of domain. For simplicity we can say a set of grains that is

G= {Barley, oats, rice, wheat, …} is a set of domain. Since when you take set G to certain function say mill you will get corresponding flour as output that is wheat flour etc.

Mathematically, if a function is defined from set X to Y then the set X is domain.

So, in the function f(x)=2x from set A to set B;

Fig: Function from A to B

The domain is X={1,2,3,4,5}

Codomain and Range:

In similar manner the Codomain is set of outputs of a function. Mathematically if we define a function from set A to B then the co-domain is set of elements of B for which there is a preimage in A. In the following figure

Fig: Function from A to B

Codomain={2,4,6,8,10}

Where as Range is set of elements in B. That is

Range={2,4,6,8,10,12}.

So, this was an article about Domain, Codomain and Range. Hope you all like this. Don’t forget to follow our blog by email to get email about what we update.

Function And its Types

The most used topic of mathematics in our life without knowing is function. So in this article we will talk about function, what is function, its types and many simple examples of function we use in our daily life.

Function And its Type

Function in simple meaning is something that gives output when you give something input. It is exactly similar to the mill when you give certain grain to input you will get flour as output. We can compare uniqueness of mill to function, that is if you put wheat to input you will get wheat flour as output you never get maze flour as output similarly in function you will get always unique output to given input.

In mathematical terms;

A relation that uniquely associates members of one set with members of another set is known as function. More formally, a function from a set X to Y is an operation f such that for every x belonging to the set X is uniquely associated with f(x) belonging to Y. Where f(x) is also an element of Y.

FIG: Function

Notation:

Normally we use f for the notation of function, or we can use letter like g or certain names also like squaring.

Examples:

• X² (square) is a function
  
•       Sine, Cosine and Tangent are some function of trigonometry.

Types of function:

Generally, there are various types of function we will discuss main type of function here:

1.     Injective function or One to one function:

A function that preserves its distinctness is known as  Injective or One to one function; it never maps distinct elements of one set namely domain to the same elements of another set namely co-domain.

Here the function is injective in since every unique element of X that is domain has unique element in Co-domain.

But here for element 3 and 4 of X has same image in Y so is not injective function. That is there must be unique image for unique element.

2.     Surjective function or Onto function

In mathematics a function from a set X to set Y is said to be Surjective or Onto function if for every element in y in Y of f there is at least one element x in X of f such that; f(x)=y. Here it is not necessary that x is unique.

Here for every element of Y there is at least one element in X so is surjective function.

3.     Bijective function:  

A function is said to be bijective function, if it is both surjective and injective. Some time it is also known as One to one correspondence.

4.     Identity function:

A function is said to be Identity function if it maps any given elements to itself. Mathematically,

      f(x)=x is Identity function.

5.     Constant function:

A function or map is said to be constant function if it maps every element of domain to same element on the codomain.

Mathematically,

f(x)=4 is constant function.

Hope you get some idea what the function is so see you in the next article.

Relation and Its Type

We are related to many peoples in our daily life, and we know what relation in daily life means so now let us talk about mathematic relations.

Relation and Its Type

Cartesian product:

A mathematical operation that returns a set from multiple sets is called Cartesian product in set theory. That is for sets A and B the Cartesian product AxB is the set of all ordered pairs (a,b) where a ɛ A and b ɛ B.

For example:

Let us take two sets A and B.

Where A= {1,2,3,4} and

              B= {a,b,c,d}

Then the cartesian product of A and B is denoted by AxB and given by:

AxB= {(1,a), (1,b), (1,c), (1,d), (2,a), (2,b), (2,c), (2,d), (3,a), (3,b), (3,c), (3,d), (4,a), (4,b), (4,c), (4,d)}

Relation:

A Relation R among sets A and B is a subset of the Cartesian product of the sets A and B.

For example:

R={(1,a), (1,b), (1,c), (2,b), (2,c) , (3,a), (3,b), (3,c)} is a relation.

A relation R among sets A and B is a subset of the Cartesian product of the sets A and B.

For example:

R={(1,a), (1,b), (1,c), (2,b), (2,c) , (3,a), (3,b), (3,c)} is a relation.

Types of Relation:

There are different types of relation we will explain about them in here:

1.     Void or empty relation:

If no element of set A is related to any element of set B, the relation in set A is called Void relation or Empty relation, . Hence R={0} which is subset of AxB.

2.     Identity Relation:

For a given set A , I={(a,a): a ɛ A} is called Identity relation in A.A is related to itself only, in identity relation.

Example:

If A={ a,b,c,d} then R={ (a,a), (b,b), (c,c), (d,d)} is identity relation in A.

3.     Symmetric Relation:

A relation R in a set A is said to be symmetric relation in A  if and only if a is related to b implies b is related to a.

Example:

The relation “equals to” is symmetric relation as a=b implies b=a.

4.     Transitive Relation:

A relation in set A is called Transitive relation if and only if a related to b and b related to c implies a related to c.

Example:

The relation “greater than” is a Transitive relation as a>b and b>c implies a>c.

5.     Reflexive Relation:

A relation R in a set A is said to be Reflexive if and only if a is related to a for all a belonging to set A.

Example:

If A={ 1,2,3,4}

Then the relation R={ (1,1), (2,2), (3,3), (4,4)} is reflexive relation.

6.     Equivalence Relation:

A relation R in a set A is said to be Equivalence relation if and only if the relation is Reflexive, Transitive and Symmetric.

So this was an article on Relation and its Type hope you all like this.