* 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 )

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

* 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:

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).

If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

If used, the word „ gothic “ was used to describe ( mostly early ) works of F. M. Dostoevsky.

If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then is provable, and then so is ¬ φ, thus φ is refutable.

If is satisfiable in a structure M, then, considering, we see that is satisfiable as well.

If the password is correct, then M releases the transferred sum to B ( 3b ), usually minus a small commission.

If M is an R module and is its ring of endomorphisms, then if and only if there is a unique idempotent e in E such that and.

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 M is an Hermitian positive-semidefinite matrix, one sometimes writes M ≥ 0 and if M is positive-definite one writes M > 0 .< ref > This may be confusing, as sometimes nonnegative matrices are also denoted in this way.

If it's also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite every eigenvalue is positive, non-negative, negative, or non-positive respectively.

If and we have for all v, w in V, then we say that B is symmetric.

If the algorithm is symmetric, the key must be known to the recipient and sender and to no one else.

If a relation is Euclidean and reflexive, it is also symmetric and transitive.

If a game has a board, the board is usually irregular rather than uniform or symmetric ( like Risk rather than chess or Scrabble ); the board is often random ( like Settlers of Catan ) or has random elements ( like Tikal ).

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

If the waves are propagating through an anisotropic ( i. e., not rotationally symmetric ) medium, for example a crystal, then the phase velocity vector and group velocity vector may point in different directions.

If G is a group and X is a set then a group action may be defined as a group homomorphism h from G to the symmetric group of X.

If confirmed, this would indicate that such symmetric fusion reactions should be modelled as ' hot fusion ' reactions rather than ' cold fusion ' ones, as first suggested.

If the quantum state is initially symmetric ( antisymmetric ), it will remain symmetric ( antisymmetric ) as the system evolves.

If the particles are bosons, they occupy a totally symmetric state, which is symmetric under the exchange of any two particle labels:

If the particles are bosons ( fermions ), the state after the measurement must remain symmetric ( antisymmetric ), i. e.

If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields ; see magnetic monopoles for details.

If the distribution is symmetric then the mean is equal to the median and the distribution will have close to zero skewness.

< li > If is a symmetric matrix of the form, and the strict inequality holds

If the problem is cylindrically symmetric, the natural solutions of the paraxial wave equation are Laguerre-Gaussian modes.

If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the shell theorem.

( If there is torsion, then the tensor is no longer symmetric.

If A is a real matrix, this is equivalent to A < sup > T </ sup > = A ( that is, A is a symmetric matrix ).

If A and B are n × n real symmetric matrices that commute, then there exists a basis of such that every element of the basis is an eigenvector for both A and B.

If A < sup >− 1 </ sup > exists, it is symmetric if and only if A is symmetric.

If Sym < sub > n </ sub > denotes the space of symmetric matrices and Skew < sub > n </ sub > the space of skew-symmetric matrices then since and

If A has real entries and is symmetric ( or more generally, has complex-valued entries and is Hermitian ) and positive definite, then A can be decomposed as