Since mathematics is related to logic, he cites an example from mathematics: If we have a formula like ( a + b )( a-b )= a²-b² it does not tell us how to think mathematically.

Since most exoplanets are in single star systems, the implicit " A " designation was simply dropped, leaving the exoplanet name with the lower-case letter only: 51 Peg b.

In the first step, the final nonzero remainder r < sub > N − 1 </ sub > is shown to divide both a and b. Since it is a common divisor, it must be less than or equal to the greatest common divisor g. In the second step, it is shown that any common divisor of a and b, including g, must divide r < sub > N − 1 </ sub >; therefore, g must be less than or equal to r < sub > N − 1 </ sub >.

Since r < sub > N − 1 </ sub > is a common divisor of a and b, r < sub > N − 1 </ sub > ≤ g.

Since his own children were born in morganatic marriage ( Sophie, Duchess of Hohenberg ), the Habsburgs designated his soon-to-be born great-nephew Robert ( b. 8 February 1915 ), second son of the future emperor Charles, as the next " adopted Austria-Este ".

Since only finitely many coefficients a < sub > i </ sub > and b < sub > j </ sub > are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Since 1983, Reform Judaism in the United States of America officially adopted a bilineal policy: one is a Jew if either of one's parents is Jewish, provided that either ( a ) one is raised as a Jew, by Reform standards, or ( b ) one engages in an appropriate act of public identification, formalizing a practice that had been common in Reform synagogues for at least a generation.

* Since m is surjective, we can find b in B such that b ′ =

* Since n is a homomorphism, n ( g ( b ) + c )

Since it is built up from the basic operations a and b, we say that the set

Since then English translations were published by many other companies ( Epic Comics, Comcat, Mojo Press, Dark Horse Comics ) resulting into all kinds of formats and quality — from b / w, American comic book sized budget collections to full color European style albums with many extras.

In field theory, the expression is only shorthand for the formal expression ab < sup >− 1 </ sup >, where b < sup >− 1 </ sup > is the multiplicative inverse of b. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero.

In June 1936 Reich wrote in his journal: " Since everything is antithetically arranged, there must be two different types of single-celled organisms: ( a ) life-destroying organisms or organisms that form through organic decay, ( b ) life-promoting organisms that form from inorganic material that comes to life.

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Since the product of an element in I with an element in R is in I ( as I is an ideal ), this product expression will be in I, and then ( a + b )< sup > n + m − 1 </ sup > is in I, therefore a + b is in the radical of I.

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a equals b. Since division by zero is undefined, the argument is invalid.

Since 1997, he was living with his partner, an artist named Jo, and their two daughters Lily ( b. 1999 ) and Saoirse.

Now suppose b < u = l is a smaller upper bound for S. Since ( l < sub > n </ sub >) is monotonic increasing it is easy to see that b < l < sub > n </ sub > for some n. But l < sub > n </ sub > is not an upper bound for S and so neither is b. Hence u is a least upper bound for S and ≤ is complete.

Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Since we can neither undo nor redo the past, we are limited to the events of today and tomorrow.

Since the recognition of Israel as a nation state, claims are made in many cases which identify the claimant as a member of the new body politic.

Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.

Since conventional methods are insensitive at the low frequencies of these molecular transitions, the paramagnetic resonance method is being used instead.

Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.

Since the provision of skywave service requires adequate freedom from interference, only class 1, stations are capable of rendering skywave service.

Since the sides are also covered up to the spray rails, they are also rough-sanded in that area.

Since fly cutters are one sided and not balanced, they should be used at the slowest speed available, and fed very slowly to avoid binding.

Since there are almost no schools in the country offering graduate work in interior design this rule cannot at present be observed.

Since charges are relatively the same, reserving a car before you leave for Europe will assure you of having one on tap when you want it.

Since practical washing procedures are both lengthy and expensive, a number of laboratory tests have been developed for the numerical evaluation of detergents.

Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.

Since emotional reactions in the higher vertebrates depend on individual experience and are aroused in man, in addition, by complex symbols, one would expect that the hypothalamus could be excited from the cortex.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.

Since N and N' are both nilpotent and they commute, the operator Af is nilpotent ; ;

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Since the choice is by lot each week, the outcomes of different trials are independent.

Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.

Since these duties fit neatly with those of the proposed presidential aide, one person, with adequate staff assistance, could fill both jobs.

Since Fogg's was a one-man, one-plane flying service, this meant that he would have to do both trips, flying alone 600 miles a day, under sub-freezing temperature conditions.

Since the adoption of a new constitution, early in 2010, the politics of Angola takes place in a framework of a presidential republic, whereby the President of Angola is both head of state and head of government, and of a multi-party system.

Since its independence, Armenia has maintained a policy of complementarism by trying to have friendly relations both with Iran, Russia, and the West, including the United States and Europe.

Since the Second World War, both theoretical and experimental fields have advanced at a rapid pace.

Since this picture was first developed by Ernst Stueckelberg, and acquired its modern form in Feynman's work, it is called the Feynman-Stueckelberg interpretation of antiparticles to honor both scientists.

Since 1972, the Roman Catholic Church uses the name " Anointing of the Sick " both in the English translations issued by the Holy See of its official documents in Latin and in the English official documents of Episcopal conferences.

Since the launch of Breakfast in 2000, the programme has been simulcast on both BBC One and BBC News, replacing the individual breakfast news programmes that had been run by both channels.

Since early 1989 the government recognizes both Israel and the Palestinian state.

Since most Western economies are now considered " open " with free capital movement, this essentially means that central banks may target interest rates or exchange rates with credibility, but not both at once.

Since then Calvin and Hobbes creator Bill Watterson has written extensively on the issue, arguing that size reduction and dropped panels reduce both the potential and freedom of a cartoonist.

Since current can be the flow of either positive or negative charges, or both, a convention for the direction of current which is independent of the type of charge carriers is needed.

Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism between the real line and the unit circle without ∞.

Since both the noun and enclitic were declined, this led to " double declension ".

Since 2002, molecular phylogenetic studies have recognized a close relationship between Hydrochoerus and Kerodon supporting placement of both genera in a subfamily of Caviidae.

Since 2001, both baronies have been redesignated the City of Dublin.

Since two houses may be very different from one another, a design pattern for houses must be broad enough to apply to both of them, but not so vague that it doesn't help the designer make decisions.

Since 2004, the colón stopped circulating and is now never used in the country for any type of transaction ; however some stores still have prices in both colons and U. S. dollars.

Since both birds are stringy, and taste unpleasant, the skin and feathers could be kept and filled with the cooked, minced and seasoned flesh of tastier birds, like goose or chicken.

Since 2002, the new Afghan government has engaged in cordial relations with both Iran and the United States, even as relations between Iran and the United States have grown strained due to American objections to Iran's nuclear program.