Mathematical induction and Boole's Law

Feb 2008
297
7
Use the additive law of probability to establish, using mathematical induction, Boole's Law:

\(\displaystyle P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})\)
 
Nov 2009
517
130
Big Red, NY
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.
 

Grandad

MHF Hall of Honor
Dec 2008
2,570
1,416
South Coast of England
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 \(\displaystyle n \ge 1\).

Grandad
 
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