There are as many Even Number as Natural Number | Introduction to Infinity

If we take Natural numbers from 2 – 10 (inclusive).

Set \  of \  numbers = \{2, 3, 4, 5, 6, 7, 8, 9, 10\}

Total  \ no. \  of \  Numbers = 9

Set \ of \ even \ number = \{2, 4, 6, 8, 10\}

Total \ no. \ of \ Even \ numbers = 5

Set \ of \ odd \ number = \{3, 5, 7, 9\}

Total \ no. \ of \ Odd \ numbers = 4

From the example above you may have thought that the total number of even or odd number is always less than that of natural number. Moreover, you may think that:

    \[ \boxed{Natural \ Number =  Even \  Numbers  +  Odd \  Numbers} \]

Which is true for finite numbers, but will you trust me if I say,

    \[ \boxed{Total \ no. \ of \ Natural \ Number = Total \ no. \ of \ Even \ Number} \]

If you think it’s some sort of joke then read the whole article. Before starting the article we need some basic concept of counting and infinity. Let’s start with counting first.

What is Counting?

Counting is simply telling how many numbers are there in the set. In the above example, Natural numbers from 2 – 10 (inclusive) we counted the number of elements of the set formed by the numbers i.e \{2, 3, 4, 5, 6, 7, 8, 9, 10\}. So, basically counting is determining the cardinal number of the set. For example:

Set \ of \ continents = \{Asia, Europe, Africa, North America, South America, Australia, Antartica\}

No. \  of \ continents = Cardinal \ Number \ of  \ set \ of \ continents = 7

Now, we know what is counting. Let’s move to another example:

A = \{2, 4, 6, 8\}

B = \{3, 5, 7, 9\}

Both set ‘A’ and set ‘B’ has the cardinality of 4 and both the set contain equal number of elements. We can demonstrate this by paring each member of set one-to-one. In the above example: the pairs are (2,3), (4,5), (6,7), (8,9).

So far we learnt what is counting and how we can say two set has equal cardinality.

If you don’t know the concept of Factorial and wonder why 0 factorial equals 1 check this,

Now let’s discuss about infinity before entering into the subject matter.

What is Infinity?

Many people got confused and think infinity as the number not just the number but the biggest number. Is it the biggest number?

Infinity is not the number but actually its the size, the size of something that doesn’t end. Infinity is not the biggest number but actually it’s how many numbers are there. There are different size of infinity. The smallest size of infinity is countable infinity and the biggest is un-countable infinity.

Countable infinity

Countable infinity is the number of whole number, the numbers of hours of forever, the number of natural numbers. Like: 1,2,3,4,5,6 …………. Sets of such numbers are unending but they are countable. Countable in a sense that you can count between two elements. For example: you can say there is 3 after 2 and 7 after 6.

UnCountable infinity

Uncountable infinity is literally uncountable. You might thought that the set of natural number is uncountable but it is not, its countable, its finite. Whats’ uncountable infinity is the number of real number. Not all the real number but the real numbers between 0 and 1 is uncountable infinity. You can’t say the next element of set after 0. You know its 0.0000….0001, but you can’t say how much 0 is there before 1.

Now you got the idea of infinity and counting lets dive into the topic.

There are as many Even Number as Natural Number

From earlier we know that to call two sets equal, the cardinal number of two sets have to be equal.

How can we say the cardinal number of two sets are equal? You think it right if they have one-to-one pair with each other.

Now lets see the example of natural number and even number:

Lets say, there are infinite number of natural number. So,

Set \  of \  Natural \ numbers = \{1,2, 3, 4, 5, 6, 7, 8, 9, 10,.............\}

Set \  of \  Even \ numbers = \{2,4,6,8,10,12,.........\}

Pairing the elements of set Natural numbers with set of Even Numbers:

Even Number as Natural Number
Even Number as Natural Number

They both have one-to-one relation.

This one-to-one relation exists for the odd number as well.

ODD Number as Natural Number
ODD Number as Natural Number

In the above two examples we saw that for infinite number of natural number the total number of natural number is equal to the number of even number as well as number of odd number. Also we know that the natural number is infinite.

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