Question on sets

Miike012

Question is in paint doc. Determine if the statement is true or false.

My solution:
I have two solutions

Sol 1: FalseIf A₁ contains A₂ and A₂ contains A₃ then the number of elements of A₃ contained in A₁ is less than the number of elements in A₂ contained in A₁. In other words the intersection of A₁ and A₃ has fewer elements than the intersection of A₁ and A₃. Therefore the intersection cannot be infinite if the elements in each consecutive intersection are decreasing.

Sol 2: True

If A₁, A₂,... A_n are sets of infinite number of elements then A₁ = A₂ = ... = A_n. For instance, how can a set containing all negative real numbers (-∞,0] and a set containing all positive real numbers [,+∞) be infinite if the set of all read numbers contains more elements than the two sets above? Therefore is it true if two or more sets have infinite elements then those sets are equal?

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clamtrox

Both of your solutions are wrong :)

False: The intersection can still be infinite

True: of course it's not true that A₁ = A₂ = ... = A_n in general.

Either you need to show that the intersection is always infinite, or find a counterexample.

vela

Neither solution is correct. You're applying your intuition from dealing with finite sets to infinite sets. That doesn't work. You can remove an infinite number of elements from an infinite set and still have an infinite number of elements remaining.

Miike012

If A1 contains An and An does not contain A1,A2...An-1 and taking in consideration that An is a set of infinite elements the intersection of A1,A2,... and An will be An which is a set of infinite elements.

Statement is True.

vela

You're misreading the problem. It's asking you about the set
$$A = \bigcap_{i=1}^\infty A_i,$$ the intersection of all of the sets. The intersection between any two of the sets will obviously contain an infinite number of elements, so it's not a very interesting question to ask.

Miike012

I understand the question. I was thinking the intersection of all the sets would be An.
If you noticed I said "the intersection of A1,A2,... and An ... "

Miike012

edit

vela

How can A_n be the intersection of all the sets? What about A_n+1?

clamtrox

A simple way to show a counterexample works is if you can show that for every [itex]x \in A_1 [/itex] there exists a [itex] n \in \mathbb{N} [/itex] such that [itex] x \notin A_n [/itex]

Miike012

I've thought of that but I can't think of an example and I can't see how that is possible if A_n is a subset of A₁

Miike012

Why would the intersection be empty? A_m and A_m+1 still have m+1, m+2, m+3,... in common, therefore the intersection of A_m and A_m+1 would be m+1, m+2, and so on... I must have the wrong interpretation of intersection.

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vela

Why do you keep talking about the intersection of two sets? As I said earlier, the question is asking about the intersection of all of the sets, i.e. ##A = A_1 \cap A_2 \cap A_3 \cap \cdots ##

clamtrox

Think again. Maybe start as easy as you can, and choose [itex]A_1 = \mathbb{N} [/itex]