Without loss of generality


Without loss of generality is a frequently used expression in mathematics. The term is used to indicate the assumption that follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are also proven by some symmetry — or another equivalence or similarity. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases.
In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property P of real numbers is known to be symmetric in x and y, namely that P is equivalent to P, then in proving that P holds for every x and y, one may assume, "without loss of generality", that x ≤ y. There is no loss of generality in this assumption, since once the case x ≤ y ⇒ P has been proved, the other case follows by y ≤ x ⇒ P ⇒ P, thereby showing that P holds for all cases.
On the other hand, if such a symmetry cannot be established, then the use of "without loss of generality" is incorrect and can amount to an instance of proof by example — a logical fallacy of proving a claim by proving a non-representative example.

Example

Consider the following theorem :
A proof:
Here, notice that the above argument works because the exact same reasoning could be applied if the alternative assumption, namely, that the first object is blue, were made. As a result, the use of "without loss of generality" is valid in this case.