# “Without loss of generality” — correct usage

When is correct to write "wlog" and what is example of wrongly used "wlog"?

For example if proving $|D(x,B) - D(y,B)| \le d(x,y)$ with $D(x,B) = \inf_{b \in B} d(x,b)$ and $(X,d)$ is a metric space is it correct to write:

"Without loss of generality assume $D(x,B) \ge D(y,B)$" so then can write $D(x,B) - D(y,B)$ instead of $|D(x,B) - D(y,B)|$?

I am not native speaker but native speaker told me wrong use is common. Thank you for help!

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I suggest that you write "without loss of generality" instead of the ugly and unpronounceable "wlog". – MJD Feb 18 '13 at 16:45
– Marc van Leeuwen Feb 18 '13 at 16:59

Your usage of "wlog" appears correct. In general, it should be used to make an artificial distinction between two or more otherwise indistinguishable things, for clarity.

For example, if we wanted to prove something about two real numbers, we might say: "let $x,y \in \mathbb{R}, x \leq y$ wlog" and then continue the proof. The point is, the numbers $x$ and $y$ are arbitrary, and one must be bigger than the other so we may as well say $x \leq y$. If someone actually gave me two points, $x = 4$ and $y = 2$ then for our proof to work we simply relabel the points so that $x \leq y$ and we have lost no generality.

On the other hand, it is often used incorrectly. Say we have a monotonic function $f : [0,1] \rightarrow \mathbb{R}$. If we wanted to prove that $f$ was injective for example, it would be wrong to start with "let $f$ be increasing wlog" because we have lost some generality in this assumption. What we should say is something like "consider the case where $f$ is increasing, and the decreasing case is similar".

It's a subtle difference, and not one that should cause too much worry to be honest, I think it's unlikely to cause any mathematical errors, it may just confuse an argument a little.

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I would second Mariano's advice. If you are not sure if you can use WLOG correctly, it is much safer to say "It suffices to consider case [specific case], because we can [describe reduction procedure]". – Willie Wong Feb 18 '13 at 16:45
@WillieWong Very true, it is probably better, but I find it a useful shorthand when you're doing a standard reduction, for example,if doing a problem involving a circle, it's enough to say for example: "wlog, pick co-ordinates such that $0$ is at the center of the circle" etc. – Tom Oldfield Feb 18 '13 at 16:49
I prefer to say, in that case, «it is enough to consider the case in which $0$ is the center of the circle, because by translation we can reduce to it from the general case» at least saying why (but not actually proving that this works, which in some cases may require quite a bit of work) – Mariano Suárez-Alvarez Feb 18 '13 at 17:04

In your example, it is correct to use it because if the inequality $D(x,B)\geq D(y,B)$ does not hold, one can simply rename the points so that $x$ becomes $y$ and $y$ becomes $x$.

«Without loss of generality» is generally used when some minor, inconsequential change in notation allows one to add an assumption. It is also used to say that «with some work, we can see that...», and that is a more annoying usage. It generally implies that the writer supposed that the reader is able to see what is going on, and that should restrict its use considerably.

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I would suggest not to use the expression, or to only use it if you immediately explain what exactly you mean by it. – Mariano Suárez-Alvarez Feb 18 '13 at 16:38