Did I assume too much in this proof?

[IntroAnalysis]

Homework Statement

Suppose F is a nonempty family of sets and B is a set. Prove that B U (∩F) = ∩_AЄF(B U A).

Homework Equations

The Attempt at a Solution

Let x Є (B U (∩F)). This means (x Є B) ν (x Є ∩F). x Є F = {x l $\forall$AЄF(xЄA)}.
Assume AЄF, then xЄA. Therefore, xЄB ν xЄA. This is equivalent to ∩_AЄF(BUA).

Let xЄ ∩_AЄF(BUA). This means xЄ{x l$\forall$A(AЄF$\rightarrow$xЄ(BUA)}. Assume AЄF, then xЄ(BUA). Thus, this is equivalent to BU(∩F).
Therefore, BU(∩F) = ∩_AЄF(BUA).

 

[mighty2000]

To prove this statement, we can use the definition of set intersection and union. The statement is essentially saying that the union of a set B and the intersection of a family of sets F is equal to the intersection of the set B with each individual set in the family F.

First, let x be an element of B U (∩F). This means that x is either an element of B or an element of every set in F. This can be written as (x Є B) ν (x Є ∩F). Using the definition of intersection, we can rewrite this as (x Є B) ν (x Є {x | ∀ AЄF (x Є A)}). This can be further rewritten as (x Є B) ν (x Є {x | ∀ AЄF (x Є B ν x Є A)}). This is equivalent to ∩AЄF(B U A), as x must be an element of both B and each set A in the family F.

Next, let x be an element of ∩AЄF(B U A). This means that x is an element of both B and each set A in the family F. Using the definition of union, we can rewrite this as x Є {x | ∀ AЄF (x Є B ν x Є A)}. This can be further rewritten as x Є {x | (x Є B) ν (x Є ∩F)}. This is equivalent to B U (∩F), as x must be an element of either B or the intersection of all sets in F.

Therefore, we have proven that B U (∩F) = ∩AЄF(B U A).