Defining Big Intersection

Hi guys.

I completely understand the definition for the intersection between two or more sets, but this definition just does not make sense.

http://upload.wikimedia.org/math/5/6...ff30b29333.png

If M is a set of sets. For all A which are an element of M, x is an element of A. Isn't that the exact same thing as the union? Am I reading this wrong? I don't see how this definition covers the fact that x HAS to appear in EVERY A. It just states that x is an element of A, not every A.

Re: Defining Big Intersection

I hope I dont say anything wrong, but this is my take on it:

You have a non empty set M. The intersection of M is the set of elements, which are in EVERY set of elements of M. Thats why there is a ∀. You just look specifially at the intersection only and not any other parts of sets. I guess thats what confuses you.

Re: Defining Big Intersection

Thanks. I was just reading the statement wrong, separating the two properties.

Re: Defining Big Intersection

Quote:

Originally Posted by

**ShadowKnight8702** http://upload.wikimedia.org/math/5/6...ff30b29333.png
If M is a set of sets. For all A which are an element of M, x is an element of A. Isn't that the exact same thing as the union? Am I reading this wrong? I don't see how this definition covers the fact that x HAS to appear in EVERY A. It just states that x is an element of A, not every A.

It is far better and preferred to say "if M is a collection of sets".

$\displaystyle \bigcap M = \left\{ {x : (\forall A \in M)\left[ {x \in A} \right]} \right\}$ and $\displaystyle \bigcup {M = \left\{ x: (\exists A \in M)\left[ {x \in A} \right]} \right\} $

Note in the first it is $\displaystyle \forall A \in M$ but in the second it is $\displaystyle \exists A \in M$.