Sentence examples for be any proper subset from inspiring English sources

Let C′ be any proper subset of C.

A cluster C of 𝒳 is any proper subset of 𝒳, explicitly excluding both the empty set ∅ and the full set 𝒳.

Both out-fragments and in-fragments are also called fragments of D. An out-fragment (in-fragment, respectively) F is minimal if any proper subset of F is no longer an out-fragment (in-fragment, respectively).

In exact deducibility the conclusion cannot be deduced from any proper subset of the premises.

Next suppose R is finite and let S be a proper subset of R with |S|⩾2.

Let (mathcal {Q}) be a proper subset of (mathcal {S}) such that (|mathcal {Q}| = alpha n).

More generally, a sufficient condition for T1 being more parsimonious than T2 is for the ontological commitments of T1 to be a proper subset of those of T2.

It is illustrated by an example that under the conditions the maximal domain of existence of a solution of (1.1) may be a proper subset of.

In other cases, however, there will be some additional restrictions on C, i.e., it will be a proper subset of the set of all maximally ordered linear subsets in (TIME,≤).

Invoke a function that for each Γ produces a subset MΓ of the models of Γ; in general, we will expect MΓ to be a proper subset of these models.

We notice that the set (class) of all well-known b-metric spaces ((bge1)) can be a proper subset (subclass) of the set (class) all transversal r-max edges space for (rge1).