If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

If we think of composition as a kind of multiplication of functions, this identity says that the inverse of a function is analogous to a multiplicative inverse.

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 is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

* Suppose & B is equivalent to & D. If we acquire new information A and then acquire further new information B, and update all probabilities each time, the updated probabilities will be the same as if we had first acquired new information C and then acquired further new information D. In view of the fact that multiplication of probabilities can be taken to be ordinary multiplication of real numbers, this becomes a functional equation

If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition.

If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring which is a topological group ( for +) in which multiplication is continuous, too.

If, as well, the multiplication is also commutative:

If A is an n × m matrix and B is an m × p matrix, the result AB of their multiplication is an n × p matrix defined only if the number of columns m in A is equal to the number of rows m in B.

If we identify with ( a, b ) then we see that addition and multiplication are given by

If we identify with ( a, b ) then we see that addition and multiplication are given by

If n is a positive integer, the integers between 1 and n − 1 which are coprime to n ( or equivalently, the congruence classes coprime to n ) form a group with multiplication modulo n as the operation ; it is denoted by Z < sub > n </ sub >< sup >×</ sup > and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, p < sup > k </ sup >, or 2 p < sup > k </ sup > where p < sup > k </ sup > is a power of an odd prime number.

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A ; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

If the multiplication is associative, an element x with a multiplicative inverse cannot be a zero divisor ( meaning for some y, xy = 0 with neither x nor y equal to zero ).

If N is a prime number, then the set of non-zero indices n = 1 ,..., N – 1 forms a group under multiplication modulo N. One consequence of the number theory of such groups is that there exists a generator of the group ( sometimes called a primitive root ), an integer g such that n = g < sup > q </ sup > ( mod N ) for any non-zero index n and for a unique q in 0 ,..., N – 2 ( forming a bijection from q to non-zero n ).

If K is not commutative, then the only change is that the order of the multiplication may be reversed, resulting in the distinct operations left scalar multiplication cv and right scalar multiplication vc.

If G is any group, then the set Ch ( G ) of these morphisms forms an abelian group under pointwise multiplication.

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

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

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state.

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.

In mathematics, and in particular linear algebra, a skew-symmetric ( or antisymmetric or antimetric ) matrix is a square matrix A whose transpose is also its negative ; that is, it satisfies the equation If the entry in the and is a < sub > ij </ sub >, i. e. then the skew symmetric condition is For example, the following matrix is skew-symmetric:

If the binary relation is antisymmetric, transitive and also total ( for all a and b in the set, aRb or bRa ), then the set is a totally ordered set.

If a tensor changes sign under exchange of any pair of its indices, then the tensor is completely ( or totally ) antisymmetric.