Without loss of generality


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 reduction to a special case

Example with analogous reasoning

Example with triviality

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


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.