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.
If A, B,..., E are the generating operations of a group, the group generated by them is represented by the symbol {A, B,..., E}.
The independent generating operations of a group may be subject to certain relations connecting them, but these must be such that it is impossible by combining them to obtain a relation expressing one operation in terms of the others.
The totality of translations constitutes, therefore, a subgroup of the general group of motions; and this subgroup is a self-conjugate subgroup, since a translation is always conjugate to a translation.
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