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1. 1. Prove: If X⊆Y then P(X)⊆P(Y).
2. Prove: P(A∩B) = P(A) ∩ P(B) with detail please
3A. Prove that If A⊆B or B⊆A then P(A∪B) = P(A) ∪ P(B)
3B. Prove #3A from the opposite direction, meaning: if P(A∪B) = P(A) ∪ P(B) then A⊆B or B⊆A
4. N is a set of natural numbers N={0,1,2....,}
For every n⊆ N => An = {x∈ N | 0≤ x ≤ n}
Prove or Disprove the following:
a) A0 = Ø
b) ∀nN An ⊆ An+1
c) n∈ N An = N
d) ∀n∈Nk∈Nm∈N |Am - An| = k
e) ∀n∈Nm∈N ((Am = {x2 | x∈An}) ↔ (m=n ^ n<2))
2. Hello, I've moved this to maths study help for you. Please post what you have attempted so far for the questions.
3. (Original post by bright_sunshine)
1. Prove: If X⊆Y then P(X)⊆P(Y)
To start you off..

By drawing a Venn diagram, we have

and

So what does that say about the required probabilities?
4. (Original post by bright_sunshine)
2. Prove: P(A∩B) = P(A) ∩ P(B) with detail please
This is meaningless - have you mistranscribed it? If so, it would make more sense to put up the original question.
5. (Original post by atsruser)
This is meaningless - have you mistranscribed it? If so, it would make more sense to put up the original question.
He/she uses the notation P(X) to stand for the power set of X, not probabilities. In either case, massive apology for you having wasted your time and efforts; the OP has posted a duplicate thread here with quite a bit of answers.

@OP: Refrain from posting duplicate threads; it wastes the time and effort of everybody trying to help you and is just impolite. Closed.

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