# Without loss of generality

https://arbital.com/p/5nr

by Jaime Sevilla Molina Jul 30 2016

Without loss of generality (abbreviated as w.l.o.g.) is a common idiom in mathematics that remarks that we can introduce a new assumption reducing the proof to a special case, and the proof for the other cases either follows from the special case, can be reasoned in an [ analogous way], or is [ trivial].

wlog is tightly related to [ case exhaustion].

## Example with triviality

Theorem: In every set of $5$ natural numbers there are three numbers which sum a multiple of $3$.

Proof:

w.l.o.g. assume that there are no three numbers with the same residue modulo $3$ in the set. Otherwise, the sum of those three numbers is a multiple of $3$.

Now, there are $5$ numbers and $3$ possible residues, so at least there is one number for each residue (otherwise, there could be a maximum of $2$ residue classes times a maximum of $2$ number per class, for a total of $4$ numbers). But $3a + (3b+1)+(3c+2) = 3 (a+b+c) + 3$, which is a multiple of $3$. Q.E.D.