Suppose that tau is an infinite set and S is a countable set

Suppose that tau is an infinite set and S is a countable set. Show that S Union tau the same cardinality as tau.

Solution

Given S is countable and T is infinite

Case 1 : T is countable

Then countable union of countable sets is countable S U T is countable.Hence S U T has the same cardinality of T

Case 2 : T is uncountable

Then Clearly S U T is uncountable. So here also S U T has the same cardinality of T