- #1
MrBeezer
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Homework Statement
Decide if the following represents a true statement about the nature of sets. If it does not, present a specific example that shows where the statement does not hold:
If A[itex]_{1}[/itex][itex]\supseteq[/itex]A[itex]_{2}[/itex][itex]\supseteq[/itex]A[itex]_{3}[/itex][itex]\supseteq[/itex]A[itex]_{4}[/itex][itex]\supseteq[/itex]...A[itex]_{n}[/itex] are all sets containing an infinite number of elements, then the intersection [itex]\bigcap[/itex][itex]^{\infty}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite as well.
Homework Equations
The Attempt at a Solution
I decided to attempt this using proof by induction.. although I'm a little unsure of how to do this using sets, here it goes anyways:
Step 1:
Show that [itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite when k=1
[itex]\bigcap[/itex][itex]^{1}_{n=1}[/itex]A[itex]_{n}[/itex]=A[itex]_{1}[/itex]
A[itex]_{1}[/itex] is infinite.
Therefore,
[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite when k=1
Next Step:
Let,
[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex] be infinite.
Inductive Step:
Show that [itex]\bigcap[/itex][itex]^{k+1}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite
[itex]\bigcap[/itex][itex]^{k+1}_{n=1}[/itex]A[itex]_{n}[/itex]=[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex][itex]\cap[/itex]A[itex]_{k+1}[/itex]
[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex][itex]\cap[/itex]A[itex]_{k+1}[/itex]=A[itex]_{k+1}[/itex]
A[itex]_{k+1}[/itex] is infinite.
Therefore,
[itex]\bigcap[/itex][itex]^{k+1}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite.
[itex]\bigcap[/itex][itex]^{\infty}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite.
Are there any glaring errors here?
Any input would be greatly appreciated.
Thanks!
-Mike