Hello acevipa Originally Posted by
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 $\displaystyle n \ge 1$.
Grandad