When can you use without loss of generality?

The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof for that case can be easily applied to all others (or that all other cases are equivalent).

Why is no loss of generality useful when deriving an equation?

(mathematics) With a constraining assumption that, however, makes it clear how to apply the proof performed under this assumption to the general case unconstrained by the assumption. Without loss of generality, assume one has the equation . …

How do you write without losing generality?

Without loss of generality, let a ≤ b ≤ c. If asked to spell this out in more detail, we might say something like: Since ≤ is a total order, the three numbers must be ordered somehow, i.e. we must have (at least) one of a ≤ b ≤ c, a ≤ c ≤ b, b ≤ a ≤ c, b ≤ c ≤ a, c ≤ a ≤ b or c ≤ b ≤ a.

How do you use WLOG proof?

Or if you’re a physicist, you prove it for x>y. Then say “by symmetry, rest is true.” And if they are equal, you can say “w.l.o.g. let x>=y”.

What is the meaning of loss of generality?

Wiktionary. without loss of generalityadverb. Making a constraining assumption that makes it clear how to apply the proof performed under this assumption to the general case unconstrained by the assumption.

What do you mean by generality?

Definition of generality 1 : the quality or state of being general. 2a : generalization sense 2. b : a vague or inadequate statement. 3 : the greatest part : bulk the generality of the population.

What is the meaning of Without loss of generality in mathematics?

Abstract. One sometimes reads in a mathematical proof that a certain assumption can be made ‘without loss of generality’ (WLOG). In other words, it is claimed that considering what first appears only a special case does nevertheless suffice to prove the general result.

How do you use generality?

Examples of generality in a Sentence He spoke in generalities as he discussed his plans for the future. I noticed the generality of the language he used in discussing his plans.

What is generality in research?

The generality of a finding refers to the degree to which a functional relationship obtained in one situation is able to predict the obtained relationship in a new situation. “Generality” refers more to functional relationships than individual events.

How do you use ‘without loss of generality’?

You can use “without loss of generality” if any situation you want to look at can be transformed to the assumption you make (in a simple way). One example could be something like proofing for any function “f” and any point “x” statement “A” holds. Then you can say “without loss of generality we can assume x=0”.

How do you prove that p(x) is true without loss of generality?

Without loss of generality, we can assume that x = z + 1 for some z ∈ Z. [In this case, S = Z and T = { z + 1: z ∈ Z } .] We want to show that P ( x) is true for all x ∈ Z. Without loss of generality, we can assume that x = 5 q + r where q, r ∈ Z and 0 ≤ r < q.

What is WLOG without loss of generality?

Without loss of generality. Jump to navigation Jump to search. Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics.

Is it possible to prove S = T?

In the above instances, indeed no generality has been lost, since in each case we can prove S = T (or, more likely, it would be assumed that the reader can deduce that S = T ). I.e., proving that P ( z) holds for z ∈ T is the same as proving that P ( x) holds for x ∈ S.