I have begun seeing the phrase "up to" a lot after taking abstract algebra. I usually can figure out what means in context.

For example, $\mathbb Z_4$ equals the set of rotations of a square, up to isomorphism.

I just don't really understand why we use it and what it actually means. The way I interpret it is sort of that, if "isomorphism" were a person, then $\mathbb Z_4$ would equal the rotations of a square, if it were up to them.

Another explanation I've seen on Wikipedia is that $A=B$, up to $x$ means that $A$ would equal $B$ if it weren't for $x$. That just doesn't make sense to me considering the first example I gave.

I hope this isn't too vague of a question, but it seems like the phrase itself is a bit vague. I guess I'm trying to find a casual understanding of how to use the phrase.

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    $\begingroup$ You can say that $\pi$ is equal to $3.14$, "up to the second digit after the decimal point," or "up to an accuracy of 0.01." It means "to within." The examples you've given are analogous to this, because two groups that are isomorphic can be considered almost the same. The difference between them is considered insignificant, in the same way that two numbers within 0.01 of one another can be considered almost the same. $\endgroup$ – Keith Jul 8 '15 at 3:49
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    $\begingroup$ possible duplicate of What does it mean for something to hold "up to isomorphism"? $\endgroup$ – user223391 Jul 8 '15 at 6:54

One way to make this rigorous (but which will unfortunately probably not quite contain all uses of the phrase) is the following:

If $\cong$ is an equivalence relation on the class $X$ of widgets then by the phrase "the set of widgets up to $\cong$" we mean either the set of equivalence classes of $X$ under $\cong$ (usually denoted $X/\cong$), or a set consisting of precisely one representative of each equivalence class (for most purposes these uses will be interchangable).

As mentioned in a comment and as you have seen on Wikipedia, however, it can also have the slightly different meaning of "if we ignore blank". In this case, it does not really make sense for blank to be an equivalence relation. But one way to reconcile them is to interpret "ignore $\cong$" as "ignore those differences that can be between objects that are equivalent under $\cong$" (this is not completely rigorous though, and as the example in the comment illustrates, there need not be some underlying equivalence relation).


We use this phrase when $A$ and $B$ not necessarily have equal elements, but there is isomorphism between them.
For example the elements of $(\mathbb Z_4,+$) are $0,1,2,3$, but the elements of square rotations, are rotations, but we can find isomorphism between them:
$0 \rightarrow 0$ degree rotation, $1\rightarrow90,2\rightarrow 180,3 \rightarrow 270$.
Hence $A=B$ "up to isomorphism" if and only if $A \cong B$


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