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I am trying to understand in what situations mathematicians use the expression "without loss of generality". The Wikipedia article states that the expression is useful in symmetric situations such as if we assume two outcomes $a$ or $b$, then we can assign a symbolic math where we can make assumptions such as assume $a>b$, so in such scenario we don't loose any generality as we can equally say assume $b>a$, because symbols are symmetric and can be used interchangeably. Is this the correct understanding of the expression?

Xander Henderson
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user0193
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3 Answers3

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Yes you are right. WLOG can be used when enforcing a particular condition can be made without really changing the nature of the problem because there is a way to transform the general case in a more particular one without impairing the solution.

Another example: to establish the formula for the radius of the circle circumscribed to three points $p_0,p_1,p_2$, we can admit WLOG that $p_0$ is the origin. Because if it is not, we can translate all three points by $-p_0$ without changing the value of the radius, and solve for the points $0,p'_1:=p_1-p_0,p'_2:=p_2-p_0$.

Another:

For the equation

$$a\cos\theta+b\sin\theta=c$$ we admit WLOG that $a^2+b^2=1$.

Because we have the equivalent equation $$\frac a{\sqrt{a^2+b^2}}\cos\theta+\frac b{\sqrt{a^2+b^2}}\sin\theta=\frac c{\sqrt{a^2+b^2}}.$$

  • Thanks for the great examples. I didnt quite catch the second example, here $a^2+b^2=1$ is the consequence of the of the last equation that you wrote, so why are we using WLOG here? whats the notion of generality here? – user0193 Sep 22 '20 at 13:17
  • @JhonnyS: do you agree that the equation is equivalent ? –  Sep 22 '20 at 13:26
  • yes, because its the parametric equation of an ellipse. – user0193 Sep 22 '20 at 13:34
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    @JhonnyS: ouch, no, it is a linear trigonometric equation, which describes the intersection of a unit circle and an straight line. But you don't need to interpret the equation to check equivalence. –  Sep 22 '20 at 13:36
  • +1 for examples. Can you please also contribute to my question, despite the fact the definition is straight forward, its just the application that makes it bit daunting. – GENIVI-LEARNER Sep 22 '20 at 13:47
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Your understanding is almost right but I wont say it is used only in symmetric cases. The expression is useful because one can define the proof using a concrete example and any such example must be broad enough to not deviate from the general sense of the proof. So the pigenhole principle example given in Wikipedia article elaborates it quite intuitively:

If three objects are each painted either red or blue, then there must be at least two objects of the same color.

A proof:

Assume, without loss of generality, that the first object is red. If either of the other two objects is red, then we are finished; if not, then the other two objects must both be blue and we are still finished.

GENIVI-LEARNER
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In addition to other answers, WLOG might be used to shorten proof or also to work with concrete notations and objetcs, instead of arbitrary ones.

For example, working in the space of continuous functions over $[a,b]$, $a,b \in \mathbb{R}$, is the same as working over the space of continuous functions over $[0,1]$, since we are able to from one to the other by the affine transformation $f(t) = a+(b-a)t$. Thus, every problem lying in the first, general space for any real bounds $a,b$, can be considered WLOG in the second one, which is fixed and easier to represent.

nicomezi
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