[SPLIT] how many closed binary operations f on A satisfy f(a

[wessleym]

1) When A = {x, a, b, c, d}, how many closed binary operations f on A satisfy f(a, b) = c?

[I wouldn't think you could assume any operation could take a and b and somehow make it c.]

2) Let A[sub1], A, B be sets with {1, 2, 3, 4, 5} = A[sub1] [c] A. B = {s, t, u, v, w, x}. f : A[sub1] -> B. If f can be extended to A in 216 ways, what is |A|?

[What does "extended" even mean here? I'm not even sure I need this question answered so much as I need to understand extensions.]

Again, I'm sorry about all these questions. But really, you have taught me very much. It's just some of the language that seems to get me down.

 

[pka]

Could a nonempty set include [null] and other integers? If it could, then wouldn't {1} and {[null],1} have the same sum?

Sorry to say this, but you show have some real confusion by that question. The ‘sum’ in the original refers to the numbers in the set. One does not add numbers to sets. Having written numerous such questions, I an convinced the you text has this one wrong. It can be done with a maximum of 21 with 7 in the set. It can be done with a maximum of 24 with 8 in the set. But cannot be done with max 24 and 7.

When A = {x,a,b,c,d}, how many closed binary operations f on A satisfy f(a,b) = c?

|AxA|=25 so there are 5<SUP>25</SUP> closed binary operations on A.
There are 5<SUP>24</SUP> closed binary operations on A that have f(a,b)=c.

Let A[sub1], A, B be sets with {1,2,3,4,5} = A[sub1] [c] A. B = {s,t,u,v,w,x}. f : A[sub1] -> B. If f can be extended to A in 216 ways, what is |A|?
[What does "extended" even mean here? I'm not even sure I need this question answered so much as I need to understand extensions.]

The number 216 is 6<SUP>3</SUP>. The number of functions from A to B is 6<SUP>|A|</SUP>. The number of functions from A<SUB>1</SUB> to B is 6<SUP>5</SUP>. Thus |A|=8.

 

[wessleym]

Okay, sorry about all the questions, then. I do believe and understand that the pigeonhole question is impossible, but I was just mentioning something I remembered from class about all sets (even nonempty sets like the one in the problem) containing the empty set as part of their contents. But I see your point.
Anyway, your other explanations are as straightforward as ever, and I understand them both. Thank you for everything. This has been quite an enlighting conversation.