Countable set?

**iamthemanyes** · February 15th 2011, 01:18 PM

Is this set countable or uncountable?

The set I of all two-element subsets of Z+
(Z+ being positive integers)

Thanks!

**Plato** · February 15th 2011, 01:38 PM

Originally Posted by iamthemanyes

Is this set countable or uncountable?
The set I of all two-element subsets of Z+ (Z+ being positive integers

Would you agree that $\mathbb{I}=\left\{ {\{ j,k\} \subset \mathbb{Z}^ + :j < k} \right\}$ is your set?

Now define $\Phi: \mathbb{I} \to \mathbb{Z}^+$ by $\{j,k\}\mapsto 2^j\cdot 3^k$ .

Can you prove that $\Phi$ is injective?

**iamthemanyes** · February 15th 2011, 01:50 PM

Fantastic! Thank you so much.

**Soroban** · February 15th 2011, 07:34 PM

Hello, iamthemanyes!

Is this set countable or uncountable?

The set $I$ of all two-element subsets of $Z^+\;\;(Z^+$ being positive integers)

Since we have a one-to-one correspondance: . $(a,b) \;\leftrightarrow\;\dfrac{a}{b}$
. . set $I$ has the same cardinality as the set of rational numbers,
. . which is countable.

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