If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.

If I now risk some comparisons with Sons And Lovers let it be clear that I am not comparing the two works or judging their merits ; ;

If nothing else, at least two good songs came out of the project, `` Out Of This World '' and `` June Comes Around Every Year ''.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.

Somebody, got to be somebody If I don't put my two cents in soon, somebody else will I know they're waitin only for one thing: for the bastards what done it to be nailed.

If we cluster together, the redcoats can make an advantage out of it, but there's not a blessed thing they can do with two or three of us except chase us, and we can outrun them ''.

If you've travelled in Europe a time or two, it is quite certain that you've had that wanting-to-be-alone feeling or that you will get it on your next visit across the Atlantic.

If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

If we try to study T using characteristic values, we are confronted with two problems.

If the distribution of the 71 items were wholly concordant in the two families, the distance would of course be 0.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If a child loses a molar at the age of two, the adjoining teeth may shift toward the empty space, thus narrowing the place intended for the permanent ones and producing a jumble.

If you are not well acquainted with the area in which you wish to locate, or if you are not sure that you and your family will like and make a success of farming, usually you would do better to rent a place for a year or two before you buy.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If we add to these contacts with friendly members the `` contacts with an organization of the church '' ( 11.2 per cent of the cases ), then a substantial two thirds of all recruitment is through friendly contact.

If the Orioles are to break their losing streak within the next two days, it will have to be at the expense of the American League champion New York Yankees, who come in here tomorrow for a night game and a single test Sunday afternoon.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

If we look about the world today, we can see clearly that there are two especially significant factors shaping the future of our civilization: science and religion.

** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.

* If two small categories are weakly equivalent, then they are equivalent.

If the demands of these two sovereigns upon his duty of allegiance come into conflict, those of the United States have the paramount authority in American law ; likewise, those of the foreign land have paramount authority in their legal system.

If there is a mixture of only two types of atoms, not counting impurities, such as a copper-nickel alloy, then it is called a binary alloy.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a tile is placed between two hotel chains of the same size, the individual player who places the tile decides which hotel chain remains on the board and which is acquired.

If two players tie for majority, they will share both shareholder bonuses.

If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

* If P is a non-empty upwards ccc poset and Y is a family of cofinal subsets of P with then there is an upwards directed set A such that A meets every element of Y.

If B is a cofinal subset of a poset A and C is a cofinal subset of B with the partial ordering of A restricted to B, then C is also a cofinal subset of A.

If a partially ordered set A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

If the condition only holds for all singleton subsets of R, then the ring is a right Rickart ring.

If S and T are subsets of a group G then their product is the subset of G defined by

If the subsets of X in Σ correspond to numbers in elementary algebra, then the two set operations union ( symbol ∪) and intersection (∩) correspond to addition and multiplication.

If X and Y are subsets of the real numbers, d < sub > 1 </ sub > and d < sub > 2 </ sub > can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, | x − y | < δ implies | f ( x ) − f ( y )| < ε.

If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X.

If X is a set, a diffeology on X is a set of maps, called plots, from open subsets of R < sup > n </ sup > ( n ≥ 0 ) to X such that the following hold:

If Y is a subset of the diffeological space X, then Y is itself a diffeological space in a natural way: the plots of Y are those plots of X whose images are subsets of Y.

If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets.

If a topological space is a Baire space then it is " large ", meaning that it is not a countable union of negligible subsets.

( Proof: If m < n, then we can view R < sup > m </ sup > as a subspace of R < sup > n </ sup >, and the non-empty open subsets of R < sup > m </ sup > are not open when considered as subsets of R < sup > n </ sup >.

If such a dichotomy theorem is true, then CSPs provide one of the largest known subsets of NP which avoids NP-intermediate problems, whose existence was demonstrated by Ladner's theorem under the assumption that P ≠ NP.

If is a topological space, is a family of subsets of which have nonempty interior such that every nonempty open set has a subset in, and is any subset of, then there is a Banach-Mazur game corresponding to.

If and are two subsets of, then

If, that is, if the two subsets have equal cardinality, then is called a balanced bipartite graph.

( If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.

If and is a family of distinct subsets of such that each subset is of size and each pair of subsets intersects, then the maximum number of sets that can be in is given by the binomial coefficient

If the set X is a topological space, then a cover C of X is a collection of subsets U < sub > α </ sub > of X whose union is the whole space X.

If there are n states, then partition the set of all finite strings into n subsets, where subset S < sub > i </ sub > is the set of strings that, when given as input to automaton A, cause it to end in state i. For every two strings x and y that belong to the same state, and for every choice of a third string z, automaton A reaches the same state on input xz as it reaches on input yz, and therefore must either accept both of the inputs xz and yz or reject both of them.

If and have additional structure, it is possible to define subsets of the set of all maps from X to Y or more generally sub-presheaves of a given presheaf and corresponding germs: some notable examples follow.

* If have a complex structure ( for instance, are subsets of complex vector spaces ), holomorphic functions between them can be defined, and therefore spaces of germs of holomorphic functions can be constructed.