MHB Prove Theorem: Probability of A Subset B is Less than B

Guest2
Messages
192
Reaction score
0
How do I prove the following theorem?

If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$

$A$ and $B$ are events and $P$ is the probability function.

What I tried (but not sure if it's right or not):

$P(B) = P((B\setminus A) \cup (B \cap A)) = P(B\setminus A)+P(B \cap A) \ge 0+P(B \cap A) \ge P(A) $

Therefore $P(B) - P(A) \ge 0$. Hence $P(B) \ge P(A)$.
 
Last edited:
Mathematics news on Phys.org
Guest said:
How do I prove the following theorem?

If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$

$A$ and $B$ are events and $P$ is the probability function.

What I tried (but not sure if it's right or not):

$P(B) = P((B\setminus A) \cup (B \cap A)) = P(B\setminus A)+P(B \cap A) \ge 0+P(B \cap A) \ge P(A) $

Therefore $P(B) - P(A) \ge 0$. Hence $P(B) \ge P(A)$.

Hi Guest! ;)

It looks right to me.
You may want to list the axioms you're using though.
There is some context missing, which is different in every textbook on the subject.
 
I like Serena said:
Hi Guest! ;)

It looks right to me.
You may want to list the axioms you're using though.
There is some context missing, which is different in every textbook on the subject.
Hi, I like Serena. :D Thanks for the reply.

I'm using the following three axioms:

Axiom 1: For every event $A$ in the class $C$,

$$P(A) \ge 0$$

Axiom 2: For the sure or certain event $S$ in the class $C$,

$$P(S) = 1$$

Axiom 3: For any number of mutually exclusive events $A_1, A_2,\cdots$, in the class C,

$$P(A_1 \cup A_2 \cup \cdots) = P(A_1)+P(A_2) +\cdots $$

How do I prove the second part of the theorem?

I know that $P(B-A) = P(B \cap A')$ but I don't know how to turn the intersection into union so that I can use axiom 3.
 
How about $B=(B-A)\cup A$?
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Back
Top