Is this set countable or uncountable?
The set I of all two-element subsets of Z+
(Z+ being positive integers)
Thanks!
Would you agree that $\displaystyle \mathbb{I}=\left\{ {\{ j,k\} \subset \mathbb{Z}^ + :j < k} \right\}$ is your set?
Now define $\displaystyle \Phi: \mathbb{I} \to \mathbb{Z}^+ $ by $\displaystyle \{j,k\}\mapsto 2^j\cdot 3^k$.
Can you prove that $\displaystyle \Phi$ is injective?
Hello, iamthemanyes!
Is this set countable or uncountable?
The set $\displaystyle I$ of all two-element subsets of $\displaystyle Z^+\;\;(Z^+$ being positive integers)
Since we have a one-to-one correspondance: .$\displaystyle (a,b) \;\leftrightarrow\;\dfrac{a}{b}$
. . set $\displaystyle I$ has the same cardinality as the set of rational numbers,
. . which is countable.