* 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 F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

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 the ranges of the morphisms of the inverse system of abelian groups ( A < sub > i </ sub >, f < sub > ij </ sub >) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j: one says that the system satisfies the Mittag-Leffler condition.

If an equation can be put into the form f ( x ) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x < sub > 1 </ sub > in the basin of attraction of x, and let x < sub > n + 1 </ sub > = f ( x < sub > n </ sub >) for n ≥ 1, and the sequence

If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β ≥ α, the point x < sub > β </ sub > lies in Y.

If Φ is a unital positive map, then for every normal element a in its domain, we have Φ ( a * a ) ≥ Φ ( a *) Φ ( a ) and Φ ( a * a ) ≥ Φ ( a ) Φ ( a *).

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.

< li > If M = ( m < sub > ij </ sub >) ≥ 0 then the diagonal entries m < sub > ii </ sub > are real and non-negative.

* If f: R → R is a Lebesgue integrable function and f ( x ) ≥ 0 almost everywhere, then

If there exists a K ≥ 1 with

If we restrict attention to real-valued W then the relation is defined only for x ≥ − 1 / e, and is double-valued on (− 1 / e, 0 ); the additional constraint W ≥ − 1 defines a single-valued function W < sub > 0 </ sub >( x ).

If n ≥ 2, then the group GL ( n, F ) is not abelian.

If is a monotone sequence of real numbers ( i. e., if a < sub > n </ sub > ≤ a < sub > n + 1 </ sub > or a < sub > n </ sub > ≥ a < sub > n + 1 </ sub > for every n ≥ 1 ), then this sequence has a finite limit if and only if the sequence is bounded.

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 the chosen anti-aliasing filter ( a low-pass filter in this case ) has a transition band of 2000 Hz, then the cut-off frequency should be ≤ 20050 Hz to yield a signal with negligible power at frequencies ≥ 22050 Hz and complete pass of frequencies ≤ 20 kHz ( within the human hearing range ).

# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).

If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β < α such that x < sub > ι </ sub > is in U for all ι ≥ β.

If X is a connected cell complex with homotopy groups π < sub > n </ sub >( X ) = 0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K ( G, 1 ) for G = π < sub > 1 </ sub >( X ).

If a function f is concave, and f ( 0 ) ≥ 0, then f is subadditive.

If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.

If the program hasn't halted yet, then it never will, since its contribution to the halting probability would affect the first N bits.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

If the expression that defines the DFT is evaluated for all integers k instead of just for, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.

If a filter is implemented using, for instance, biquad stages using op-amps, N / 2 stages will be needed.

# If the remainder from dividing N by 6 is not 2 or 3 then the list is simply all even numbers followed by all odd numbers ≤ N

If f: M → N is any function, then we have f id < sub > M </ sub >

If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately N < nowiki >!</ nowiki >.

If the molecule is symmetrical, e. g. N < sub > 2 </ sub >, the band is not observed in the IR spectrum, but only in the Raman spectrum.

# If M, N ∈ Λ, then ( M N ) ∈ Λ

If the alphabet consists of alternative symbols, each symbol represents a message consisting of N bits.

If μ is a positive measure, then N is null ( or zero measure ) if its measure μ ( N ) is zero.

If μ is not a positive measure, then N is μ-null if N is | μ |- null, where | μ | is the total variation of μ ; equivalently, if every measurable subset A of N satisfies μ ( A )