* Let H be a group, and let G be the direct product H × H. Then the subgroups

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).

Let f and g be any two elements of G. By virtue of the definition of G, = and =, so that =.

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Let a, b, and c be elements of G. Then:

Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.

Let R be a ring and G be a monoid.

Let this point be called G. Point G is also the midpoint of line segment FQ:

Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).

Let ( G ,.

Let G and H be groups, and let φ: G → H be a homomorphism.

Let G be a group.

Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

Let G be a group.

Let N and K be normal subgroups of G, with

Let Af denote the form of Af.

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

Let X denote a Cauchy distributed random variable.

Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude

Let denote the equivalence class to which a belongs.

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set.

Let R denote the field of real numbers.

Let n denote a complete set of ( discrete ) quantum numbers for specifying single-particle states ( for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.

Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

Let denote the space of scoring functions.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Let us denote the time at which it is decided that the compromise occurred as T.

Let denote the sequence of convergents to the continued fraction for.

Let us denote the mutually orthogonal single-particle states by and so on.

That is, Alice has one half, a, and Bob has the other half, b. Let c denote the qubit Alice wishes to transmit to Bob.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

Let Q ( x ) denote the number of square-free ( quadratfrei ) integers between 1 and x.

A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates.

Let π < sub > 2 </ sub >( x ) denote the number of primes p ≤ x such that p + 2 is also prime.

Let be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.

Let me set the record this time, and let me get back OK, so the German will give me the exclusive.

Let him bounce back, and he could really set up the staff.

Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

* 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 )

Let A be a set, and apply the axiom of regularity to

Most recently Chuck D became involved in Let Freedom Sing: The Music of the Civil Rights, a 3-CD box set from Time Life.

He's also set to appear in a follow up movie called Let Freedom Sing: The Music That Inspired the Civil Rights Movement.

* Let X be a simply ordered set endowed with the order topology.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

Let X be a nonempty set, and let.

Let X be a finite set with n elements.

* Let be the set of ordered pairs of integers with not zero, and define an equivalence relation on according to which if and only if.

* Let be an arbitrary set, and let be the set of all bijective functions from to.

The 2005 Kazuo Ishiguro novel Never Let Me Go and the 2010 film adaption are set in an alternate history in which cloned humans are created for the sole purpose of providing organ donations to naturally born humans.

Let ƒ be a function whose domain is the set X, and whose range is the set Y.

Let ( S ,*) be a set S with a binary operation * on it ( known as a magma ).

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.