# Without loss of generality

Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics. The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof for that case can be easily applied to all others, or that all other cases are equivalent or similar.[1] Thus, given a proof of the conclusion in the special case, it is trivial to adapt it to prove the conclusion in all other cases.

This often requires the presence of symmetry. For example, in proving ${\displaystyle P(x,y)}$ (i.e., that some property ${\displaystyle P}$ holds for any two real numbers ${\displaystyle x}$ and ${\displaystyle y}$), if we wish to assume "without loss of generality" that ${\displaystyle x\leq y}$, then it is required that ${\displaystyle P}$ be symmetrical in ${\displaystyle x}$ and ${\displaystyle y}$, namely that ${\displaystyle P(x,y)}$ is equivalent to ${\displaystyle P(y,x)}$. There is then no loss of generality in assuming ${\displaystyle x\leq y}$, since a proof for that case can trivially be adapted for the other case ${\displaystyle (y\leq x)}$ by interchanging ${\displaystyle x}$ and ${\displaystyle y}$ (leading to the conclusion ${\displaystyle P(y,x)}$, which is known to be equivalent to ${\displaystyle P(x,y)}$, the desired conclusion.)

## Example

Consider the following theorem (which is a case of the pigeonhole principle):

A proof: Template:Bquote

This works because exactly the same reasoning (with "red" and "blue" interchanged) could be applied if the alternative assumption were made, namely that the first object is blue.