# Prove that real numbers and irrational numbers are numerically equivalent. [duplicate]

How do I prove that set A and set A/C are numerically equivalent,where A is uncountable set and C is countably finite subset of A. This can be further used to show that real numbers and irrational numbers are numerically equivalent.
I am aware of the fact that,what are uncountable and countably infinite sets and also the meaning of numerically equivalent.I am not able to create the required bijection.

## marked as duplicate by Brian M. Scott, ervx, marty cohen, Asaf Karagila♦ elementary-set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 10 '16 at 15:14

Choose a countable subset $D$ of the irrationals, say all the rational translates of $\pi$. Then form a 1-1 correspondence between $D$ and $D\cup C$ where $C$ are the rationals. Together with the identity on $\mathbb{R}\setminus(C\cup D)$ this provides the desired correspondence.