** 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 S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* 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 M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

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

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If Af is the operator induced on Af by T, then evidently Af, because by definition Af is 0 on the subspace Af.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).

If T is a good symmetry of the dynamics, then

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.

If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.

If they are right, they will prevail of and by themselves.

Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If we break the minister to our bit, we are buying back our own sins.

If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.

If Daddy's books are out of bounds his own picture books are not.

If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.

If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??

If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.

If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.

If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.

If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.

If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.

If the UN troops are not mercenaries then the Hessians were not mercenaries either.

If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.

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

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.