Solution to Measure, Integration & Real Analysis by Axler Section 2A Exercise 2A3

Prove that if $A,B\subset\mathbf R$ and $|A|<\infty$, then $$|B\setminus A|\geqslant |B|-|A|.$$

Solution: Note that we have $$B\subset A\cup (B\setminus A).$$By 2.8, we have $$|B|\leqslant |A|+|B\setminus A|.$$Since $|A|<\infty$, we can subtract both sides by $|A|$ and hence obtain $$|B\setminus A|\geqslant |B|-|A|.$$

Remark: The condition $|A|<\infty$ is necessary for subtracting. For example, $2 > 1$, but $$2-\infty = 1 – \infty.$$ Namely, subtracting $\infty$ does not preserve the inequality.