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Thursday, February 2, 2012

The integers and the natural numbers have the same cardinality

One could argue that there are twice as many integers as there are natural numbers since for every natural number there are two integers: $1 \mapsto (1,-1)$, $2 \mapsto (2,-2)$.

You can imagine that there was quite some opposition from within the mathematics community when Georg Cantor (1845-1918) proposed the following theorem:
Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.

A sequence for $\mathbf{N}$ is $$s(n) = \sum_{k=1}^n 1 \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 4 \\
\cdots & \cdots \end{array} $$

Likewise a sequence for $\mathbf{Z}$ is: $$s(n) = \sum_{k=1}^n (-1)^{k+1} \cdot k \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & -1 \\
3 & 2 \\
4 & -2 \\
\cdots & \cdots \end{array} \, $$ We can thus establish a bijective (one-to-one) map between $\mathbf{N}$ and $\mathbf{Z}$.

By the theorem above we can conclude that $\mathbf{N}$ and $\mathbf{Z}$ have the same cardinality ( 'number of elements' ).

1 comment:

  1. Wonderful blog and nice explanation,I am here to discuss something about natural numbers.Natural numbers is the set of non-negative integers and this includes zero as a natural number.


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