# Math Help - proof of axioms

1. ## proof of axioms

given that set B is ccontained in a set A, show that the probability of A is greater than or equal to that of B

2. Short proof: All the elements in B are also in A.

Long proof:

$\displaystyle B \subseteq A$

$\displaystyle n(B) \leq n(A)$

$\displaystyle \frac{n(B)}{n(\varepsilon)} \leq \frac{n(A)}{n(\varepsilon)}$

$\displaystyle Pr(B) \leq Pr(A)$.

3. Originally Posted by Prove It
Short proof: All the elements in B are also in A.

Long proof:

$\displaystyle B \subseteq A$

$\displaystyle n(B) \leq n(A)$

$\displaystyle \frac{n(B)}{n(\varepsilon)} \leq \frac{n(A)}{n(\varepsilon)}$

$\displaystyle Pr(B) \leq Pr(A)$.
What is n(.) and what rules of proof are in force here?

Also re thread title: You can't proove axioms (unless there is redundancy in the set of axioms but we do try to avoid that).

Come to think of it do you not have to assume that both probabilities exist?

CB

4. Originally Posted by CaptainBlack
What is n(.) and what rules of proof are in force here?

Also re thread title: You can't proove axioms (unless there is redundancy in the set of axioms but we do try to avoid that).

Come to think of it do you not have to assume that both probabilities exist?

CB
The n(A) notation stands for number of elements in A.

5. Originally Posted by Prove It
The n(A) notation stands for number of elements in A.
Are you then assuming finite sets and equally likely cases, I'm not sure your argument works even with countably infinite sets.

You also leave too much notation undefined.

CB

6. Originally Posted by CaptainBlack
Are you then assuming finite sets and equally likely cases, I'm not sure your argument works even with countably infinite sets.

You also leave too much notation undefined.

CB
I disagree, this is standard notation - even year 8 students should be aware of the notation of $\displaystyle \varepsilon$ to represent the universal set, $\displaystyle n(A)$ to represent the number of elements in set A, and that $\displaystyle Pr(A) = \frac{n(A)}{n(\varepsilon)}$. Even if they are infinite, the same logic holds if you were to draw a Venn Diagram of the situation.

7. Originally Posted by Prove It
I disagree, this is standard notation - even year 8 students should be aware of the notation of $\displaystyle \varepsilon$ to represent the universal set,
Ha ha ha ... you jest

$\displaystyle n(A)$ to represent the number of elements in set A,
Cardinality? That is not the notation I know.

and that $\displaystyle Pr(A) = \frac{n(A)}{n(\varepsilon)}$. Even if they are infinite, the same logic holds if you were to draw a Venn Diagram of the situation.
Let your $\varepsilon$ be the unit disk, let A be some disk contained in $\varepsilon$, now what does your notation mean?

CB

8. Originally Posted by omoboye
given that set B is ccontained in a set A, show that the probability of A is greater than or equal to that of B
The title did say “from the axioms”
Recall that $\left( {\forall C} \right)\left[ {P(C) \geqslant 0} \right]$ and $C \cap D = \emptyset \; \Rightarrow \;P(C \cup D) = P(C) + P(D)$.

Use those. $A = B \cup \left( {A\backslash B} \right)$, therefore $P(A) = P(B) + P\left( {A\backslash B} \right) \geqslant P(B)$.