maya_jai_singh
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Hey,

Was doing some research when I encountered a definition on a^b which states that this is the number of sets of b elements which can be chosen from a set of a elements. So 2^1 = 2 because the number of sets of 1 element which can be chosen from a set of 2 elements is 2. But then if we think of 2^2, this is defined as the number of sets of 2 elements that can be chosen from a set of 2 elements. Surely this is just 1? But then 2^2 = 4 so I'm confused how the definition works in this case.

Sorry if I'm being really silly, below is the website I am using for reference.

https://brilliant.org/wiki/what-is-00/

Any help would be greatly appreciated, thank you so much!
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qweqworiet
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Think about the order you are choosing them
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maya_jai_singh
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(Original post by qweqworiet)
Think about the order you are choosing them
So, if we have a set of 2 elements call them A, B and we need to find the number of way of choosing 2 elements from this set then surely we can have A, B or B,A hence 2^2 = 2? But how is that equal to 4?

So sorry if I'm being silly once again.
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qweqworiet
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What about AA and BB?
Are you sure they weren’t talking about aCb?
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ghostwalker
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(Original post by maya_jai_singh)
Hey,

Was doing some research when I encountered a definition on a^b which states that this is the number of sets of b elements which can be chosen from a set of a elements. So 2^1 = 2 because the number of sets of 1 element which can be chosen from a set of 2 elements is 2. But then if we think of 2^2, this is defined as the number of sets of 2 elements that can be chosen from a set of 2 elements. Surely this is just 1? But then 2^2 = 4 so I'm confused how the definition works in this case.

Sorry if I'm being really silly, below is the website I am using for reference.

https://brilliant.org/wiki/what-is-00/

Any help would be greatly appreciated, thank you so much!
Can't access that website as you need to log in.

Really need to see the original material you've got that definition from, as the terminology you've used is not consistent with that definition which may be what's causing your issue. As you say, there is only one set of 2 elements where those elements are taken from a set of 2 elements.
Last edited by ghostwalker; 1 month ago
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DFranklin
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(Original post by maya_jai_singh)
Sorry if I'm being really silly, below is the website I am using for reference.

https://brilliant.org/wiki/what-is-00/
I'd be careful about what sites you trust. I think that definition is wrong.

It would be better to think of m^n as the number of ordered n-tuples (an n-tuple is just an ordered set of n elements) with elements 1,2, ..., m. e.g. when m = 2, n = 2, you'd have (1, 1), (1, 2), (2, 1), (2, 2).

There's some somewhat more formal discussion here: https://en.wikipedia.org/wiki/Exponentiation#Over_sets

(Original post by ghostwalker)
Can't access that website as you need to log in.

Really need to see the original material you've got that definition from, as the terminology you've used is not consistent with that definition which may be what's causing your issue. As you say, there is only one set of 2 elements where those elements are taken from a set of 2 elements.
The wording on the site (I didn't need to log in) is:

"0^0 represents the empty product (the number of sets of 0 elements that can be chosen from a set of 0 elements), which by definition is 1. This is also the same reason why anything else raised to the power of 0 is 1."

Which to my mind is unambiguously wrong (I can't really find a way of looking at it that isn't wrong). Edit: I guess you can weasel out of "wrong" by saying "chosen works here and there was no implication it worked for numbers larger than 0".
Last edited by DFranklin; 1 month ago
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ghostwalker
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(Original post by DFranklin)
The wording on the site (I didn't need to log in) is:
Got in now - by clicking outside the login overlay. Ta.
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qweqworiet
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So it is not talking about any a^b but only ^0
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maya_jai_singh
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(Original post by DFranklin)
I'd be careful about what sites you trust. I think that definition is wrong.

It would be better to think of m^n as the number of ordered n-tuples (an n-tuple is just an ordered set of n elements) with elements 1,2, ..., m. e.g. when m = 2, n = 2, you'd have (1, 1), (1, 2), (2, 1), (2, 2).

There's some somewhat more formal discussion here: https://en.wikipedia.org/wiki/Exponentiation#Over_sets


The wording on the site (I didn't need to log in) is:

"0^0 represents the empty product (the number of sets of 0 elements that can be chosen from a set of 0 elements), which by definition is 1. This is also the same reason why anything else raised to the power of 0 is 1."

Which to my mind is unambiguously wrong (I can't really find a way of looking at it that isn't wrong). Edit: I guess you can weasel out of "wrong" by saying "chosen works here and there was no implication it worked for numbers larger than 0".
This makes so much more sense to me! Thank you so much. So for 0^0, this would be defined as the number of ordered set of 0 elements with element 0? But if we have 0 elements in a set, we cannot have the element 0 in that set. So would 0^0 = 1 hold for the definition you have mentioned?

Thanks so much once again though for taking out your time, means a lot.

So if we had 0^0 this would be the number of ordered 0-tuples with elements 0 but then surely this is 0 as we cannot have 0 sets of 0 elements?
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maya_jai_singh
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(Original post by qweqworiet)
So it is not talking about any a^b but only ^0
Ahh ok, thank you so much for helping me out!
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maya_jai_singh
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(Original post by ghostwalker)
Can't access that website as you need to log in.

Really need to see the original material you've got that definition from, as the terminology you've used is not consistent with that definition which may be what's causing your issue. As you say, there is only one set of 2 elements where those elements are taken from a set of 2 elements.
Yeah you're right, I do think the definition caught me out. Cheers, means a lot!
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DFranklin
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(Original post by maya_jai_singh)
This makes so much more sense to me! Thank you so much. So for 0^0, this would be defined as the number of ordered set of 0 elements with element 0? But if we have 0 elements in a set, we cannot have the element 0 in that set. So would 0^0 = 1 hold for the definition you have mentioned?

Thanks so much once again though for taking out your time, means a lot.

So if we had 0^0 this would be the number of ordered 0-tuples with elements 0 but then surely this is 0 as we cannot have 0 sets of 0 elements?
The empty set has no elements. So there is one set with no elements.

Edit: if you're wanting arguments for "why might we want 0^0 = 1", there's an argument similar to the binomial theorem,, but much more general. It is very common to define a general polynomial p(x) in the form p(x) = \sum_0^n a_n x^n, which is (implicitly) saying that a_0 x^0 = a_0 for all choices of x including x = 0. (and so assumes 0^0 = 1).
Last edited by DFranklin; 1 month ago
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