# Mathematical induction and Boole's Law

#### acevipa

Use the additive law of probability to establish, using mathematical induction, Boole's Law:

$$\displaystyle P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})$$

#### Anonymous1

Use the additive law of probability to establish, using mathematical induction, Boole's Law:

$$\displaystyle P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})$$
Common proofs like this one can be found online.
Here is one:

http://www.andrew.cmu.edu/course/21-228/lec7.pdf

There are many others that may better suit your needs, though.

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Hello acevipa
Use the additive law of probability to establish, using mathematical induction, Boole's Law:

$$\displaystyle P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})$$
Since you are asked to use induction, the proof will be something like this:

Suppose that the proposition is true for $$\displaystyle n = k$$. So
$$\displaystyle P(A_{1}\cup A_{2}\cup...\cup A_{k})\leq P(A_{1}) + P(A_{2})+...+P(A_{k})$$
Then
$$\displaystyle P\Big((A_{1}\cup A_{2}\cup...\cup A_{k})\cup A_{k+1}\Big)$$ $$\displaystyle =P(A_{1}\cup A_{2}\cup...\cup A_{k})+P(A_{k+1})-P\Big((A_{1}\cup A_{2}\cup...\cup A_{k})\cap A_{k+1}\Big)$$, using the addition law of probability
$$\displaystyle \leq P(A_{1}\cup A_{2}\cup...\cup A_{k})+P(A_{k+1})$$, since $$\displaystyle P\Big((A_{1}\cup A_{2}\cup...\cup A_{k})\cap A_{k+1}\Big) \geq 0$$
$$\displaystyle \Rightarrow P(A_{1}\cup A_{2}\cup...\cup A_{k}\cup A_{k+1})$$ $$\displaystyle \leq P(A_{1}) + P(A_{2})+...+P(A_{k})+P(A_{k+1})$$, using the Induction Hypothesis

When $$\displaystyle n = 1$$, the hypothesis is clearly true. So it is true for all $$\displaystyle n \ge 1$$.