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.
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.
Some regard without any loss of generality (WALOG for short) as a more grammatically correct expression.
WLOG is invoked in situations where some property of a model or system is invariant under the particular choice of instance attributes, but for the sake of demonstration, these attributes must be fixed.
WLOG can also be invoked to shorten proofs where there are a number of choices of configuration, but the proof is “the same” for each of them.
For example, the proof of the fundamental theorem of arithmetic uses this notion, in essence settling on a “canonical form” for prime factorizations to simplify the argument.
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