* 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 a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.

If a diatomic molecule consists of two atoms of the same element, such as H < sub > 2 </ sub > and O < sub > 2 </ sub >, then it is said to be homonuclear, but otherwise it is heteronuclear.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

* If we think of as the ring of real numbers, then the direct product again consists of.

* If is the set of all cars, and is the equivalence relation " has the same color as ", then one particular equivalence class consists of all green cars.

If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function such that:

If the domain consists of the non-negative numbers, then the function is injective and invertible.

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

If either party consists of two or more members, all will come to attention, but only the most senior member of the party will offer ( or return ) the physical or verbal salute.

If the diad consists of two identically oriented units, the diad is called a meso diad reflecting similar features as a meso compound.

If the diad consists of units oriented in opposition, the diad is called a racemo diad as in a racemic compound.

If the array of weights consists of N weights spaced evenly over the length L

If the fully qualified domain name of any name server for a zone appears within that zone, the zone administrator provides IP addresses for that name server, which are installed in the parent zone as glue records ; otherwise, the delegation consists of the list of NS records for that zone.

If absent or deficient of natural tissue, alternatives can be cultured sheets of epithelial cells in vitro or synthetic compounds, such as integra, which consists of silicone and bovine tendon collagen with glycosaminoglycans.

If one assumes that all of the universe consists of protons and neutrons, the density of the universe is such that much of the currently observed deuterium would have been burned into helium-4.

If ( E, M ) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense.

* If K is a field and we consider the K-vector space K < sup > n </ sup >, then the endomorphism ring of K < sup > n </ sup > which consists of all K-linear maps from K < sup > n </ sup > to K < sup > n </ sup >.

If the body consists of a homogeneous material with sufficiently known physical properties, the thermal mass is simply the mass of material present times the specific heat capacity of that material.

If the support of a distribution consists of points, with corresponding probabilities, then the probability mass function of the distribution over can be written, using the Kronecker delta, as

If ecological sanitation is practiced municipal wastewater consists only of greywater, which can be recycled for gardening.

If a thermodynamical system consists of N particles, then a point in the 6N-dimensional phase space describes the dynamical state of every particle in that system, as each particle is associated with three position variables and three momentum variables.

If virtue consists in being useful to our fellow citizens, perhaps there were few more virtuous men than Mr. Craig.

If the universe of discourse consists of all functions from the real line R to itself, then the solutions for f are all functions whose only values are one and zero.

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

If a stranger entered the circle, he had to do so to the woman's right so as not to come between her and her husband.

* If a circle passing through two of the input points doesn't contain any other of them in its interior, then the segment connecting the two points is an edge of a Delaunay triangulation of the given points.

If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse.

Formulae connecting a tangential angle, the angle anchored at the ellipse's center ( called also the polar angle from the ellipse center ), and the parametric angle t < ref > If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle is the geocentric latitude, and parametric angle t is a parametric ( or reduced ) latitude of auxiliary circle </ ref > are:

If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are ( more or less ) Kac – Moody algebras.

If the circle is replaced by an ellipse, and rotated about the major axis, the shape becomes a prolate spheroid, rotated about the minor axis, an oblate spheroid.

If a particular point on a sphere is ( arbitrarily ) designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them.

If one measures by arc length one finds that the shortest path connecting two points lying entirely in the sphere is a segment of the great circle containing the points ; see geodesic.

If an object moves with angular velocity ω around a circle of radius r centered at the origin of the x-y plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

( If one of the basic hand-shape glyphs is used, such as the simple square or circle, this band breaks it in two ; however, if there are lines for fingers extended from the base, then they become detached from the base, but the base itself remains intact.

If the handle is circular with the stem as the axis of rotation in the center of the circle, then the handle is called a handwheel.

If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function ( of integer order ) of the radial component.

If we expand a function f in a power series inside a circle of radius R, this means that

If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the " string " is constrained between the adjacent arcs of the cycloid, and the pendulum's length is equal to that of half the arc length of the cycloid ( i. e. twice the diameter of the generating circle ), the bob of the pendulum also traces a cycloid path.

If the Earth were a perfect sphere and there were no atmosphere, the radio horizon would be a circle.

If we take to be a circle, where < math > r <

If there is no radial component, then the particle moves in a circle.

If modeled on the practice of using dictionaries, there would be no circle in an illustration of the activity of looking up a word whose entry provides a definition in terms of that word ( or in terms of another word defined in terms of this word ).

If conditions are completely clear, the circle is empty.

If conditions are partly cloudy, the circle is partially filled in.

If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit.

If your motion is not strictly in a line or a circle your hand will describe an ellipse and the wave will be elliptically polarized.

If they conceive of the circle casting as cutting a line in the air with the tool of air, then they may choose to purify the circle with the remaining three elements of fire, water, and earth ; this would involve using a candle to purify the circle, and omitting the incense, since the circle has already been imbued with the element of air.