How Do Sets {B1, B2, ..., Bn} Form an n-Partition of Set G?

[occhi]

Homework Statement

Hi i can't understand the question i need to help show me how to reach a solution

Let {A₁,A₂...A_n} be an n-partition of the space $$\Omega$$,and define the sets{B₁,B₂,...} by
B_j=G$$\cap$$A_j,j=1,2,3... .Show that the sets {B₁,B₂...B_n} is an n-partition of the set G

Homework Equations

The Attempt at a Solution

 

[jbsteele]

Let {A1, A2, ... , An} be an n-partition of the space Ω, i.e. A1 ∪ A2 ∪ ... ∪ An = Ω and A1 ∩ A2 ∩ ... ∩ An = Ø. Let G be any subset of Ω. Since the sets {A1, A2, ... , An} form a partition of Ω, we can define the sets {B1, B2, ... , Bn} by Bi = G ∩ Ai, where i = 1, 2, ... , n. We will now show that the sets {B1, B2, ... , Bn} form a partition of G. To show this, we must show that B1 ∪ B2 ∪ ... ∪ Bn = G and B1 ∩ B2 ∩ ... ∩ Bn = Ø. We first show that B1 ∪ B2 ∪ ... ∪ Bn = G. Since G ⊆ Ω, it follows that G ∩ Ai ⊆ Ai for each Ai in the partition. Therefore, B1 ∪ B2 ∪ ... ∪ Bn ⊆ A1 ∪ A2 ∪ ... ∪ An = Ω. Since G ⊆ Ω and G ∩ Ai ⊆ Ai for each Ai in the partition, it follows that G = (G ∩ A1) ∪ (G ∩ A2) ∪ ... ∪ (G ∩ An). This implies that G = B1 ∪ B2 ∪ ... ∪ Bn. We now show that B1 ∩ B2 ∩ ... ∩ Bn = Ø. Since A1 ∩ A2 ∩ ... ∩ An = Ø, it follows that G ∩ A1 ∩ A2 ∩ ... ∩ An = Ø. Therefore, B1 ∩ B2 ∩ ... ∩ Bn = (G ∩ A1) ∩ (G ∩ A