  In this paper, the concepts of set-valued anti-homomorphism and strong
set-valued anti-homomorphism of $\Gamma$-semigroup are introduced. The notions
of generalized lower and upper approximation operators, constructed by means of
set-valued mapping, which is a generalization of the notion of lower and upper
approximation of $\Gamma$-semigroup, are provided. Some significant properties
of lower and upper approximations of $\Gamma$-semigroup under
(strong)set-valued anti-homomorphisms are discussed. We provide examples for
the properties of lower and upper approximations of $\Gamma$-semigroup under
(strong)set-valued anti-homomorphisms. This article explores the notion of a
set-valued anti-homomorphism for $\Gamma$-semigroups, which is a generalization
of anti-homomorphism in general
