- #1
soumyashant
- 9
- 0
Can you prove that [tex]\mathbf{R}[/tex] and [tex]\mathbf{R}-\mathbf{Q}[/tex] have same cardinality?
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.
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.