Can you prove that [tex]\mathbf{R}[/tex] and [tex]\mathbf{R}-\mathbf{Q}[/tex] have same cardinality?(adsbygoogle = window.adsbygoogle || []).push({});

One way would be to say that [tex]\mathbf{R}-\mathbf{Q}[/tex] is not countable and must have cardinality <= [tex]\mathbf{R}[/tex] and invoke the Continuum Hypothesis to conclude that its cardinality is aleph-1 same as that of [tex]\mathbf{R}[/tex]..

Somehow this does not look appealing...

Can you explicitly construct a bijection and help me to visualise the situation better??

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Construction of bijection

**Physics Forums | Science Articles, Homework Help, Discussion**