Prove this number fact Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$. 
Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought $\frac{0}{0}$ was undefined. 

If xy=0, then x=0 or y=0.  So, by contraposition, if is not the case that "x=0 OR y=0", then it is not the case that xy=0.  By De Morgan's Laws, we have that "if it is not the case that x=0 AND it is not the case that y=0, then it is not the case that xy=0".  Now we move the negations around as a "quantifier exchange" and we have "if x is not equal to 0 and y is not equal to 0, then xy is not equal to 0."  
If xy=0, we can't say that xy/xy=1.  Division doesn't exist in the rational numbers.  It only exists in non-zero rational numbers, or if you prohibit 0 in any denominator.  xy/xy=1 only on the condition that "/" is defined.  So this "Suppose xy=0. Then xy/xy=1." isn't correct.
