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

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: