The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

The horizontal line y = c is a horizontal asymptote of the function y = ƒ ( x ) if

Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where ( x, y, z ) belongs to the subset if and only if f ( x, y ) = z.

Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.

Antimony is stable in air at room temperature, but reacts with oxygen if heated to form antimony trioxide, Sb < sub > 2 </ sub > O < sub > 3 </ sub >.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

The automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

The color of amethyst has been demonstrated to result from substitution by irradiation of trivalent iron ( Fe < sup > 3 +</ sup >) for silicon in the structure, in the presence of trace elements of large ionic radius, and, to a certain extent, the amethyst color can naturally result from displacement of transition elements even if the iron concentration is low.

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

After the indecisive < ref name =" British historian Townsend Miller "> British historian Townsend Miller: “ But, if the outcome of < nowiki > battle of </ nowiki > Toro, militarily, is debatable, there is no doubt whatsoever as to its enormous psychological and political effects ” in The battle of Toro, 1476, in History Today, volume 14, 1964, p. 270 </ ref > Battle of Toro in 1476 against King Ferdinand II of Aragon, the husband of Isabella I of Castile, he went to France to obtain the assistance of Louis XI, but finding himself deceived by the French monarch, he returned to Portugal in 1477 in very low spirits.

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

His comment on Numbers 23: 19 has a still more polemical tone: “ God is not a man that he should lie ; neither the son of man, that he should repent ; < font face =" times new roman " size = 3 > if a man says: ‘ I am a god ’ he is a liar ; if he says: ‘ I am a son of man ’ he will have cause to regret it ; and if he says, ‘ I will go up to heaven ’ he has said but will not keep his word ” last phrase is borrowed from B ' midbar 23: 19 ( Yer.

In the statement < code >< u > if </ u > token = if < u > then </ u > < u > result </ u >

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

A function f ( x ) is asymptotic to the straight line ( m ≠ 0 ) if

A convex function | function is convex if and only if its Epigraph ( mathematics ) | epigraph, the region ( in green ) above its graph of a function | graph ( in blue ), is a convex set.

It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length | r | can be constructed with compass and straightedge.

Image: pons_asinorum. png | The bridge of asses theorem states that if A = B then C = D.

Image: Thales ' Theorem Simple. svg | Thales ' theorem: if AC is a diameter, then the angle at B is a right angle.

More formally, if a, b, c, and d are all integers, then the fraction < sup > a </ sup >⁄< sub > b </ sub > is irreducible if and only if there is no other equal fraction < sup > c </ sup >⁄< sub > d </ sub > such that | c | < | a | or | d | < | b |, where | a | means the absolute value of a.

More importantly, several of the more advanced of the less developed countries have found through experience that they must plan their own complex investment programs for at least 5 years forward and tentatively for considerably more than that if they are to be sure that the various interdependent activities involved are all to take place in the proper sequence.

For example, the marksman gets 5 shots, but we take his score to be the number of shots before his first bull's-eye, that is, 0, 1, 2, 3, 4 ( or 5, if he gets no bull's-eye ).

One method of accomplishing this moor is to set a bow anchor normally, then drop back to the limit of the bow cable ( or to double the desired scope, e. g. 8: 1 if the eventual scope should be 4: 1, 10: 1 if the eventual scope should be 5: 1, etc.

Many stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to cast shadows if they were only 10 parsecs from the Earth: Rigel (− 7. 0 ), Deneb (− 7. 2 ), Naos (− 6. 0 ), and Betelgeuse (− 5. 6 ).

However, if this pixel uses premultiplied alpha, all of the RGB values ( 0, 1, 0 ) are multiplied by 0. 5 and then the alpha is appended to the end to yield ( 0, 0. 5, 0, 0. 5 ).

For example, if the array has five elements, indexed 1 through 5, and the base address B is replaced by B − 30c, then the indices of those same elements will be 31 to 35.

Eco ( 1993 ) notes that Genesis is ambiguous on whether the language of Adam was preserved by Adam's descendants until the confusion of tongues ( Genesis 11: 1-9 ), or if it began to evolve naturally even before Babel ( Genesis 10: 5 ).

The anode should be removed and checked after 5 years ( sooner if there is a sodium based water softner inline ), and replaced if 15 cm ( 6 inches ) or more of bare wire is showing.

And even this 1. 5 % will be reduced to 1 %, if we take out India and China, which have not pursued liberal trade and industrial policies recommended by the developed countries.

5 ) be concurrent, it does not follow that points in a portion of a plane perpendicular at O to the axis will be also concurrent, even if the part of the plane be very small.

The percentage of fluent speakers turns out to be even higher if those under 16 are also taken into account, given that the proportion of bilinguals is particularly high in this age group ( 76. 7 % of those aged between 10 and 14 and 72. 4 % of those aged 5 – 9 ): 37. 5 % of the population aged 6 and above in the whole Basque Autonomous Community, 25. 0 % in Álava, 31. 3 % in Biscay and 53. 3 % in Gipuzkoa.

It was once fatal before the CT era, now, if the abscess is treated before the person goes into a coma, then the death rate has been estimated from 5 % to 20 % although it is greater in cases of multiple abscesses, when raised intracranial pressure is observed and depending on the level of neurological dysfunction on presentation.

For example if a player rolls a 6 and a 5, but has no checkers on the 6-point, though 2 checkers remain on the 5-point, then the 6 and the 5 must be used to bear off the 2 checkers from the 5-point.

Assuming that 10 – 15 % of bread is thrown away as stale, if it could be made to stay fresh another 5 – 7 days then perhaps 2 million tons of flour per year would be saved.

According to a 2010 interview on Blog Talk Radios, Lessons Learned, Rick Tocquigny, when asked if Mumy was a Jonathan Harris fan, before Mumy's first meeting with Harris on Lost in Space, he said at age 5, he was too young to watch his mentor's show The Third Man which was probably late at night, but was old enough to watch The Bill Dana Show ( which also starred Harris's real-life best friend Don Adams ).

For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1 in 6 chance of any single number appearing, assuming that you get the original amount wagered back.

Both the team and its fans ( in the poem, about 5, 000 attended the game ) believe they can win " if only " they could somehow get " Mighty Casey " ( Mudville's star player ) up to bat.

And if anyone saves a life, it would be as if he saved the life of all people " ( Qur ' an 5: 32 ).