Without loss of generality
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Without loss of generality or simply WLOG is a frequently used expression in mathematics. The term is generally used where there is some kind of symmetry that allows the situation or situations described to be trivially generalized to all needed situations.
Example
Consider the following theorem (the simplest case of Ramsey's theorem and also an example of Dirichlet's pigeonhole principle):
Three objects are each painted either red or blue; there must be two objects of the same color.
The proof:
- Assume without loss of generality that the first object is red. If either of the other two objects is red, we are finished; if not, the other two objects must both be blue and we are still finished.
We can assume WLOG that the first object is red, because there is no difference between red and blue for the purposes of the proof. If the first object is blue instead of red, that is equivalent to a mere change of the names of the two colors, and the names of the colors don't matter; the proof goes through just fine if you switch 'red' to 'blue' and vice versa.