Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where ( x, y, z ) belongs to the subset if and only if f ( x, y ) = z.

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

A binary operation is a binary function where the sets X, Y, and Z are all equal ; binary operations are often used to define algebraic structures.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; sometimes, the more general ' states ' are used instead of levels, to avoid using degeneracy in the equation ), N is the total number of particles and Z ( T ) is the partition function.

The following is a typical reaction scheme, where C represents the catalyst, X and Y are reactants, and Z is the product of the reaction of X and Y:

For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.

* In stem-based definition, A refers to the most inclusive clade containing X, Y, etc., and their common ancestor, down to where Z branches off below A. Taxa are included between the node of A and down to ( but not including ) the branching point to Z ; that is, the stem of A.

* Z ( where ω is a cube root of 1 ), the ring of Eisenstein integers.

Two isomorphic constructions of the field with 4 elements are ( Z / 2Z )/( T < sup > 2 </ sup >+ T + 1 ) and Z /( 2Z ), where φ =.

This phenomenon, where molecule Y affects the binding of molecule X to a transport molecule Z, is called a heterotropic allosteric effect.

Some drummers even use completely mismatched hi-hats from different cymbal ranges ( Zildjian's K / Z hats ), of different manufacturers, and even of different sizes ( similar to the K Custom Session Hats where the top hat is a sixteenth of an inch smaller than the bottom ).

Also consider the group ( Z < sub > 2 </ sub > × Z < sub > 3 </ sub >, +), the ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the x-coordinate is modulo 2 and addition in the y-coordinate is modulo 3.

where Z < sub > p </ sub > denotes the p-adic integers.

HBC Rewards / Club Z is a large program first started by Zellers in 1986 as Club Z. Loblaws offers the President's Choice Financial program where cardholders earn PC Points towards free groceries.

An example of a ternary relation ( i. e., between three individuals ) is: " X < tt > was-introduced-to </ tt > Y < tt > by </ tt > Z ", where ( X, Y, Z ) is a 3-tuple of persons ; for example, " Beatrice Wood was introduced to Henri-Pierre Roché by Marcel Duchamp " is true, while " Karl Marx was introduced to Friedrich Engels by Queen Victoria " is false.

One prominent example is the anime Maziger Z, where the term " Super Robot ", features in the Japanese theme song According to Go Nagai:

where n < sup > c </ sup > denotes the charge conjugate state, i. e., the antiparticle.

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where r is the position vector of the particle relative to the origin, p is the linear momentum of the particle, and × denotes the cross product.

where and denotes the portfolio value at time t.

respectively, where e < sub > x </ sub >, e < sub > y </ sub >, e < sub > z </ sub > denotes the cartesian basis vectors ( all are orthogonal unit vectors ) and A < sub > x </ sub >, A < sub > y </ sub >, A < sub > z </ sub > are the corresponding coordinates, in the x, y, z directions.

where denotes the complex conjugate of.

denotes the rank-one operator that maps the ket to the ket ( where is a scalar multiplying the vector ).

Just as kets and bras can be transformed into each other ( making into ) the element from the dual space corresponding with is where A < sup >†</ sup > denotes the Hermitian conjugate ( or adjoint ) of the operator A.

where C < sub > α </ sub > denotes I < sub > α </ sub > or e < sup > απi </ sup > K < sub > α </ sub >.

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.

where denotes the length of a string p.

where J denotes the Jacobian matrix of partial derivatives of ƒ and is the induced norm on the matrix.

for every automorphism φ of G ( where φ ( H ) denotes the image of H under φ ).

where denotes the vector cross product and square brackets denote evaluation in the rotating frame of reference.

This functional relationship is often denoted y = f ( x ), where f denotes the function.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

where the star denotes complex conjugation.

A *( B *( C * D )) where A, B, C, D are partially transparent image layers and "*" denotes a compositing operator ( with the left layer on top of the right layer ).

With this view it turns out that the extractor property is equivalent to: for any source of randomness that gives bits with min-entropy, the distribution is-close to, where denotes the uniform distribution on.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

where denotes the largest integer that

where ζ denotes the Riemann zeta function ( see Lehmer ; one approach to prove the inequality is to obtain the Fourier series for the polynomials B < sub > n </ sub >).

where denotes the largest integer that is not greater than x.

where denotes Newton's gravitational constant,