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December 22nd, 2016, 04:12 PM   #1
Joined: Oct 2016
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Math Focus: Still deciding!
If A is a set.. sequences of elements in A

What I know:
If A is a set and $\displaystyle {x_n}$ is a sequence of elements of A then it's not necessarily true that the terms $\displaystyle x_1, x_2, x_3,...$ are distinct.

What I'm confused on:
The proof I am reading has a statement, "Arrange the elements of $\displaystyle x$ of A in a sequence $\displaystyle {x_n}$ of distinct elements."

So I get that what they are basically saying is that we are putting the elements of A into sequence, in other words, loosely speaking we are giving each of them a number.

My question is, are we assuming that all the elements of A are distinct? Do we assume that the phrase, "Let A be a set" means that they are distinct elements? If they are not distinct, it doesn't matter right? We still just make a sequence of all of the terms but we would treat the repeated elements as if they are distinct by giving them their own number. (By number I mean their own $x_n$ for some n).

Sorry if this is an "obvious question" but I coudn't find a clear answer in any of the books I have. I am attaching a picture of the proof I am reading.
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File Type: jpg 15621747_354409601600954_5568039160211942921_n.jpg (41.6 KB, 5 views)

Last edited by ProofOfALifetime; December 22nd, 2016 at 04:15 PM.
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December 22nd, 2016, 04:46 PM   #2
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Okay, so after I posted this I found the answer. We assume that they are distinct because $\displaystyle {2,3, 2, 3, 4, 5, ...}$would not be a valid set. Sorry everyone.
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February 4th, 2017, 04:40 PM   #3
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The sentence "Arrange the elements x of A in a sequence xn of distinct elements"
doesn't say that a general sequence has to have distinct elements, just that we can write all the elements of x in such a sequence. (This is true only for countable sets.)
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