Question

how to prove ., if P(AUB)=P(A)UP(B) then (B part of A
or A part OF B)

Answer 1

what does P present?

Answer 2

P = if A={x,y,z} , P(A)={{x},{y},{z},{xy},{xz},{yz},{xyz},{}}

Answer 3

p(A) U p(B)=p(A) + P(B) - P(AnB)

Answer 4

\(\mathcal P(A\cup B)=\mathcal P(A) \cup \mathcal{P}(B) \implies [A\subset B \ \vee \ B\subset A] \)

Answer 5

since P(A) = Probabililty of only A + prob of A and B

Answer 6

Is this set theory or statistics?

Answer 7

no clue :P

Answer 8

lol, im no good with stats, but would "part of" have any significance?

Answer 9

i gett the question now okay so u gotta move this A C B or B C A

Answer 10

prove*

Answer 11

you can prove

if P(A U B ) = P(A) U P(B) , then A intersect B = empty set

Answer 12

Well he gave an example of what he meant which was he is meaning P to be the power set .
Suppose P(A U B)=P(A) U P(B) .
Then {x} is in P(A U B) and P(A) U P(B). This also means that x is A U B.
Then continue from here.

Answer 13

it is not true in general that P(A U B ) is a subset of P(A) U P(B)
therefore it is false that P( A U B ) = P(A) U P(B)

Answer 14

let A = {x} , B = {y}

P(AUB) = P( {x} U { y } ) = P ( {x , y }) = { 0, {x}, {y}, {x,y} }

but


P(A) U P(B) = P ( {x} ) U P({y} ) = { 0, {x }} U { 0, {y}} = { {x} , {y} , 0 }


so it is false that

P(A U B ) is a subset of P(A) U P(B)

so they can't be equal. because if A = B , then A < B and B < A

Answer 15

i mean , that if the = is true , if is it really : P( A U B ) = P(A) U P(B)
then it must be [(A subseteq B )or (B subseteq A)]\]
( i know that my English is bad , but what i can do , I', try'n my best to improve my self in every thing)

Answer 16

@perl we get to suppose the if part

Answer 17

correct

Answer 18

the if part is P(AUB)=P(A)UP(B)

Answer 19

I'm sorry. Maybe I missed something.

Answer 20

basically i showed that your claim is false

Answer 21

it is false that
P( A U B ) = P(A) U P(B)

and I constructed a counterexample

Answer 22

no no... We get to suppose P(AUB)=P(A)UP(B)

Answer 23

the if part is P(AUB)=P(A)UP(B)

Answer 24

oh

Answer 25

I know P(AUB) is not equal to P(A)U P(B) in general

Answer 26

ok

Answer 27

ok we want to prove
if P(AUB)=P(A)UP(B) then B < A or A < B

Answer 28

But we have IF P(AUB)=P(A)UP(B) THEN B subset of A or A subset of B.

Answer 29

maybe a proof by contradiction

Answer 30

perl that would be a cute way

Answer 31

It is probably the most easiest way if I'm not mistaken.

Answer 32

but, you need to start with A and B

Answer 33

https://www.physicsforums.com/threads/may-someone-perform-a-quick-check-of-this-proof-involving-disjunctions.565589/

Answer 34

it seems to be the answer , thank you @perl @freckles