** 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 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 a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

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 he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

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

# If A is Lebesgue measurable, then so is its complement.

If the measured value does not lie in B, replace B by its complement for the above state.

If all those bits are Equal to zero, skip the next instruction ; and in any case, replace those bits by their boolean complement.

# Σ is closed under complementation: If A is in Σ, then so is its complement,

If a universe U is defined, then the relative complement of A in U is called the absolute complement ( or simply complement ) of A, and is denoted by A < sup > c </ sup > or sometimes A ′, also the same set often is denoted by < MATH >

If L is recursively enumerable, then the complement of L is recursively enumerable if and only if L is also recursive.

If a normal operator on a finite-dimensional real or complex Hilbert space ( inner product space ) stabilizes a subspace, then it also stabilizes its orthogonal complement.

If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples ( such as the complement of a trefoil knot ) where a manifold may have a finite volume geometric structure of either type.

The only answer to that possible today is this: If the Russian Revolution becomes the signal for a proletarian revolution in the West, so that both complement each other, the present Russian common ownership of land may serve as the starting point for a communist development ".

If one takes an " open set " to be a set whose complement is closed, then the family of all open sets forms a topology.

If A is a recursive set then the complement of A is a recursive set.

If an open set V lies in the causal complement of an open set U, then the image of the maps

A family F of graphs is said to be closed under the operation of taking minors if every minor of a graph in F also belongs to F. If F is a minor-closed family, then let S be the set of graphs that are not in F ( the complement of F ).

If the subtrahend has fewer digits than the minuend, leading zeros must be added which will become leading nines when the complement is taken.

* If two's complement representation is used, subtraction requires only inverting the bits of the subtrahend and setting a carry into the rightmost bit.

They replied: " If the Russian Revolution becomes the signal for a proletarian revolution in the West, so that both complement each other, the present Russian common ownership of land may serve as the starting point for a communist development.

If M is a positive-definite symmetric matrix, then so is the Schur complement of D in M.

If the complement is not finite, but it is countable, then one says the set is cocountable.

** If a simple game is proper and strong, then a coalition is winning if and only if its complement is losing, that is, iff.

If G is a cycle of five vertices, it is isomorphic to its complement, but this property is not true for longer odd cycles, and it is not as trivial to compute the clique number and chromatic number in an odd antihole as it is in an odd hole.

If there is doubt about the cause of the skin lesions, a biopsy of the skin may be performed to distinguish the purpura from other diseases that cause it, such as vasculitis due to cryoglobulinemia ; on microscopy the appearances are of a hypersensitivity vasculitis, and immunofluorescence demonstrates IgA and C3 ( a protein of the complement system ) in the blood vessel wall.

If the complement is finite, then one says Y is cofinite.