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In this paper, we introduce the notions of (∈, ∈∨q_δ^k)-fuzzy subnear-ring and (∈, ∈∨q_δ^k)-fuzzy ideal of a near-ring which are generalization of (∈, ∈∨q_k)-fuzzy subnear-ring and (∈, ∈∨q_k)-fuzzy ideal respectively. We provide the characterizations of (∈, ∈∨q_δ^k)-fuzzy subnear-ring (resp. ideal) of a near-ring and deal with some related properties. Further, we introduce the notion of ∈∨q_δ^k-level subset of a fuzzy subset and obtain some related results.