Without loss of generality
Encyclopedia
Without loss of generality (abbreviated to WLOG; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics
. The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof on this subset can be easily applied to all others (or that all other cases are trivial). Thus, given a proof of the special case, it is trivial to show that the conclusions follow from the full premise.
This often requires the presence of symmetry. For example, if two numbers are called x, y, and it is known that x < y, then any relationship proved based on this assumption will hold for the complementary relation, y < x, because the roles of x and y are interchanged, but the proof is symmetric in the two variables. In other words, if we know that P(x, y) is true if and only if
P(y, x) is true, then without loss of generality it is enough to show P(x, y) is true (since P(y, x) then immediately follows, by symmetry). (In this context, we call P symmetric
.)
(which is a case of the Pigeonhole Principle):
A proof:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof on this subset can be easily applied to all others (or that all other cases are trivial). Thus, given a proof of the special case, it is trivial to show that the conclusions follow from the full premise.
This often requires the presence of symmetry. For example, if two numbers are called x, y, and it is known that x < y, then any relationship proved based on this assumption will hold for the complementary relation, y < x, because the roles of x and y are interchanged, but the proof is symmetric in the two variables. In other words, if we know that P(x, y) is true if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
P(y, x) is true, then without loss of generality it is enough to show P(x, y) is true (since P(y, x) then immediately follows, by symmetry). (In this context, we call P symmetric
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
.)
Example
Consider the following theoremTheorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
(which is a case of the Pigeonhole Principle):
A proof: