Let [{[[phi].sub.i] : [X.sub.i] [right arrow] X}.sub.i[member of][GAMMA]] be a family of

injective functions and [{([[B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (E, M)-dffb's on [X.sub.i] satisfying the following condition:

Related to this last result, we observe that the functions constructed in the proof are in general discontinuous (since we cannot have a continuous

injective function [PHI] : [R.sup.n] [right arrow] R for n [greater than or equal to] 2).

Now by the preserver property of [phi] and Lemma 7, it follows that there is an

injective function [f.sub.1] : [0,1] [right arrow] R such that [f.sub.1](tr [phi](P)[phi](Q)) = [f.sub.1](tr PQ) which implies that

Since f is an intuitionistic fuzzy

injective function, A [intersection] [f.sup.-1](B) = [0.sub.~] and [f.sup.-1](B) is an intuitionistic fuzzy [G.sub.2str] clopen set in Y containing [y.sub.r,s].

--P is a finite, nonempty set of productions of the form P = ([Sigma], [Tau] [Phi]), where [Sigma] is the left-hand side, a subgraph whose nodes are labeled with symbols of N; [Tau] is the right-hand side, a subgraph whose nodes are labeled with symbols of N; and [Phi] is the inheritance function, where [Phi]: [Tau] [right arrow] [Sigma] is a partial,

injective function that indicates the nodes of [Tau] that will inherit the connecting edges of the nodes of [Sigma]

The public key generator chooses two groups [G.sub.1] and [G.sub.2] of prime order p, two independent generators g, [??], a bilinear map e:[G.sub.1] x [G.sub.1] [right arrow] [G.sub.2], and an

injective function [mu]:GF(p) x {1, ..., n} [right arrow] GF(q) (for example, [mu](x, y) = (y - 1)p + x [18]) and a collision-resistant hash function H(x).

Then, there exists an

injective function [THETA] from the set of linear extensions of O to the set of linear extensions of [O.sub.up] and furthermore, [THETA] is surjective if and only if O = [O.sub.up] or O = [O.sub.down].

Graham and Sloane [1] defined a (p, q)-graph G of order p and size q to be harmonious, if there is an

injective function f : V(G) [right arrow] [Z.sub.q], where [Z.sub.q] is the group of integers modulo q, such that the induced function [f.sup.*] : E(G) [right arrow] [Z.sub.q], defined by [f.sup.*] (xy) = f(x) + f(y) for each edge xy [member of] E(G), is a bijection.

Let now h : N [right arrow] G be an

injective function. For U [subset or equal to] G we define the lower density dens [inf.sub.h] U, upper density dens [sup.sub.h] U, and density (if it exists) [dens.sub.h] U as those of the preimage [h.sup.-1](U).

An

injective function f : V (G) [right arrow] {[l.sub.0],[l.sub.1],[l.sub.2],...,[l.sub.q]}, is said to be strong Lucas graceful labeling if the induced edge labeling [f.sub.1](uv) = |f(u) ?

* : U x U [right arrow] U is an

injective function,

Therefore, we may define an

injective function [xi], m = [xi](n) (Figure 7) so that the inverse range of 0 by H is given by