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Claim: 2 is irrational.

Proof of the claim: Suppose2 is rational, i.e., 2 = a/b for some integers a & b with b >0.

We can assume that b is positive, since otherwise we can simply change the signs of both a & b.

(Then a is positive too, although we will not need this).Let us choose integers a & b with 2 = a/b, such that b is positive and as small as possible. (Wecan do this by the Well-Ordering Principle, which says that every nonempty set of positiveintegers has a smallest element).

Squaring both sides of the equation 2 = a/b and multiplying both side by b2, we obtain a2 = 2b2.

Since a2 is even, it follows that a is even. Thus, a = 2kfor some integer k, so a2 = 4k2, and henceb

2 = 2k2. Since b2 is even, it follows that b is even. Since a & b are both even, a/2 and b/2 areintegers with b/2 > 0, and 2 = (a/2)/ (b/2), because (a/2)/ (b/2) = a/b.

But we said before that b is as small as possible, so this is a contradiction. Therefore, 2 cannotbe rational.

This particular type of proof by contradiction is known as innite descent, which is used to provevarious theorems in classical number theory. If there exist positive integers a and b such that a/b= 2, then the above proof shows that we can nd smaller positive integers a and b with thesame property, and repeating this process, we will get an innite descending sequence of positiveintegers, which is impossible.

Recall that in the above proof, we said we can assume that b>0, since otherwise we can simply

change the signs of both a & b. Other way to write this would be Without Loss OfGenerality (WLOG), b > 0.

Without Loss Of Generality (WLOG) means that there are two or more cases (in this proofthe cases when b > 0 and b < 0), but considering just one particular case is enough to prove thetheorem, because the proof for the other case or cases works the same way.

In essence, therefore, the phrase WLOG may be used to indicate that the time and effort savingact of consideration of any one of the many possibilities (e.g. b>0 and b

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A similar practice is prevalent within the domain of legal civil procedure. Here, the wordPrejudice refers to a loss or injury, and refers specifically to a formal determination inrespect of a claimed legal right orcause of action. Thus, when a rule made under a certain actof law says The Central Government shall have power to make rules in respect of all or any ofthe matters referred to in section 3. In particular, and Without Prejudice to the generality ofthe foregoing power, such rules may provide for all or any of the following matters, namely: (a)the standards of quality of air, water or soil for various areas and purposes; (b) maximumallowable limits of concentration of various environmental pollutants for different areas; (c) theprocedures and safe guards for handling of hazardous substances; ,etc the intention behind

usage of the phrase Without Prejudice is to prevent any loss or injury to the generality ofthe foregoing power by way of avoiding any restriction which can prevent enforcement of certainmeasures which may be necessary, in certain specific circumstances, for the purpose ofprotection and improvement of the quality of environment and prevention, control and abatementof environmental pollution.

Thus, the rephrased rule may be conceived as The Central Government shall have power tomake rules in respect of all or any of the matters referred to in section 3. In particular, andWLOG of the foregoing power, such rules may provide for all or any of the following matters,namely: (a) the standards of quality of air, water or soil for various areas and purposes; (b)

maximum allowable limits of concentration of various environmental pollutants for differentareas; (c) the procedures and safe guards for handling of hazardous substances; , etc.

Most of us are introduced to "without loss of generality" before encountering formal grouptheory. To the uninitiated, the phrase almost seems like cheating, but soon we realize howintuitive and useful it is for simplifying and shortening proofs.

As hinted above, is it true that behind every WLOG there is an implied symmetry group in play?

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Some Examples

Schurs Inequality: For nonnegative x, y, z and for r > 0 one has the symmetric inequality:

0 xr(x y)(x z) + yr(y x)(y z) + zr(z x)(z y)

(We have equality here if and only if either (a) all three variables are equal or (b) a pair of thevariables are equal and the third is zero).

Proof:

WLOG, assume x y z. We can do this because the expression at the right hand is symmetricin x, y, z. The symmetry group is S3 .

The Fundamental Theorem of Algebra is a theorem about equation solving. It states that

every polynomial equation over the field of complex numbers of degree higher than 1 has a

complex solution. Polynomial equations are of the form:

P(x) = anxn + an-1x

n-1 + ... + a1x + a0 = 0

Where, an is assumed non-zero in which case n is called the degree of the polynomial P and of

the equation above. ai's are known coefficients while x is an unknown number. A numbera is a

solution to the equation P(x) = 0 if substituting a for x renders it identity: P(a) = 0. Thecoefficients are assumed to belong to a specific set of numbers where we also seek a solution.

The symmetry group, in this case, is is the multiplicative group of non-zreo real numbers.

The Pigeonhole Principle: If three objects are to be painted in two colors, red and blue, then

there must be two objects of the same color.

Proof:

Assume WLOG 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. The symmetrygroup is.