# set operations

• Mar 13th 2013, 01:44 AM
baxy77bax
set operations
hi,

can anyone show me why is this true:

$\displaystyle P(\cap_{i=1}^{n}E_{i}) \leq P(\cup_{i=1}^{n}E_{i})$

P is a probability function

thank you
• Mar 13th 2013, 03:44 AM
Plato
Re: set operations
Quote:

Originally Posted by baxy77bax
can anyone show me why is this true:
$\displaystyle P(\cap_{i=1}^{n}E_{i}) \leq P(\cup_{i=1}^{n}E_{i})$
P is a probability function

You should have proved that if $\displaystyle A~\&~B$ are events and $\displaystyle A\subseteq B$ then $\displaystyle \mathcal{P}(A)\le\mathcal{P}(B)$.

Simply apply that here.
• Mar 13th 2013, 05:32 AM
baxy77bax
Re: set operations
ok i get it , however i still don't get how to show that

$\displaystyle \cap E \subset \cup E$

or is this an axiom
• Mar 13th 2013, 05:47 AM
Plato
Re: set operations
Quote:

Originally Posted by baxy77bax
ok i get it , however i still don't get how to show that
$\displaystyle \cap E \subset \cup E$ or is this an axiom

There is no point in your trying to prove any theorem in probability if you are not grounded in set theory.

Surely you can show that $\displaystyle A\cap B\subseteq A\cup B~?$
• Mar 21st 2013, 05:19 AM
HallsofIvy
Re: set operations
Do you understand the definitions of "$\displaystyle \cap$" and "$\displaystyle \cup$"? $\displaystyle x\in A\cap B$ if and only if x is in both A and B. $\displaystyle x\in A\cup B$ if x is in either A or B.

If $\displaystyle x\in \cap E$, then x is in every set in E. So certainly, $\displaystyle x\in \cup E$.