Sentence examples for an equivalent relation on from inspiring English sources

It is clear that the relation ~ is an equivalent relation on I. Denote by I /~ as the quotient set, and I s 's, which is called index sets of ζ, as the elements of I/~.

Therefore, D defines an equivalent relation on (mathbb{R}times Xtimes X), where the equivalent class of ((t,a,b)) is [toverrightarrow{ab}] = bigl{ soverrightarrow{uv} : tlangle overrightarrow{ab},overrightarrow{xy}rangle= slangleoverrightarrow {uv}, overrightarrow{xy}rangle, forall x, y in X bigr}.

Define a relation '∼' in L by x ∼ y if and only if x ∧ a = y ∧ a for all x, y ∈ L. Then we can see that the relation ∼ is an equivalent relation on L. Definition 4.6 Let L be a lattice and I = 〈 a 〉 be a principal ideal of L. We call I a semi-standard ideal if it satisfies the following condition: x ∼ y implies ( x ∨ b ) ∼ ( y ∨ b ) for all b ∈ I.

In addition, this paper mainly focuses on the case that the fractional order is (0<alpha<2), since there exists an equivalent relation of fractional-order systems with order (0 <andhaleq1) and with order (1leqbeta< 2); see [15].

In Section 3, we construct an equivalent relation between exponential dichotomy on time scales and the admissibility of the pair ((mathrm{C}_{mathrm{rd}}^{b}(mathbb{T}^, X),L^{p}(mathbb{T}^,X))) for the evolution family on time scales.

Meanwhile, an equivalent relation between temperature load and vapor concentration load is presented.

We can easily see that this relation is an equivalent relation.

In this section, we give an equivalent relation between KKMP and SFBFP.

Also, they involve the group adjoint representation which establishes an equivalent relation among all conjugate sub-algebra elements.

An equivalent relation has been derived for a static configuration network model with an arbitrary degree distribution (Miller 2012).

It is equivalent to give a groupoid over a basis X or an equivalence relation on X together with one group for each equivalence class.