Prove that *BIG'* = *BIG* - *Little* (set difference) is uncountable. Let BIG be an uncountable set and let Little be a countable one. Prove that BIG' = BIG - Little (set difference) is uncountable.
 A: Suppose BIG' is countable then BIG=BIG'$ \cup$ Little thus BIG is countable, a contradiction. 
A: If $Little$ is countable, there is a 1-1 mapping between it and the integers, indeed between it and just the odd integers.  Now suppose $BIG'$ is countable then must be a mapping between it and the integers, or indeed just the even integers.  Now $BIG = BIG' \cup Little$.   So take $x\in BIG$, if $x\in$ little, map it to an odd integer.  If instead it is in $BIG'$, map it to an even integer.  
Now we have constructed a 1-1 mapping between $BIG$ and the integers, and have proved that $BIG$ is countable.  This contradicts the assumption that $BIG$ was uncountable.  So our supposition that $BIG'$ is countable must be false.  Thus $BIG'$ is uncountable.
(Note, lots of things in the above proof need to be cleaned up.  For example I have confused "countable" with "countable and infinite").
A: Suppose that $BIG-Little$ were countable.  If we can show that the union of two countable sets is countable, then we will have a contradiction since $(BIG - Little) \cup Little = BIG$.
Take two countable sets $A$ = $\{a_0, a_1, a_2, \dots \}$ and $B = \{b_0, b_1, b_2, \dots \}$. We can make a 1-1 correspondence $f: \mathbb{N} \to A\cup B$ by defining $f(n) =  a_{n/2}$ for even $n$, and $f(n) = b_{(n-1)/2}$.  A set with a  1-1 correspondence with $\mathbb{N}$ is countable by definition. 
