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Measure on difference of two sets


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.

Linearity

This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.

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