Help plz ! we say that 1) #x\inuuu_(n\inN)A_n# if #x\inA_n# for some #n\inN# and 2)#x\innnn_(n\inN)A_n# if #x\inA_n# for all #n\inN# prove or dispprove #nnn_(n\inN)A_n\sube uuu_(n\inN)A_n# for any index set #N# ?

1 Answer
KillerBunny
Jun 21, 2018

True.

Explanation:

Let's call the intersection of all sets #A# and their union #B#.

We want to prove that #A# is a subset of #B#. By definition, this happens when every element of #A# is also an element of #B#, whereas the contrary is not necessarily true.

So, we need to show that

#x \in A \implies x \in B#

We only need to recall the definition, and everything gets really simple: if we assume that #x \in A#, it means that #x \in A_n# for all #n \in N#. On the other hand, in order to #x \in B#, it is sufficient that #x \in A_n# for at least one #n \in N#.

So, if we assume that #x# belongs to all the sets #A_n#, is it true that it belongs to at least one of the #A_n#? Well, of course it is.

On the other hand, if we know that #x# belongs to at least one of the #A_n#, can we conclude that it belongs to all the #A_n#? Not necessarily.

This is enough to prove that the set given by the intersection is a subset of the set given by the union.

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