HOW MANY SUBSETS WHICH THEIR OPERATIONS WITH A FIXED SUBSET CONTAIN AT LEAST ONE ELEMENT OF A GIVEN COLLECTION?

We pose the following problem related to binary set operations on finite sets. Given a finite set . Let a binary set operation and be a non-empty collection of non-empty subsets of . For a fixed subset of , where , how many subsets of which their operation with contains at least one element of ?. In this paper, we give the solution of this problem, especially for the subsets of size , using the inclusion-exclusion principle, Corrádi’s lemma, and Bonferroni’s inequality. In this context, the problem is related to determining the degree of nodes in certain graphs, such as graphs constructed with the adjacency rule depends on and the node set is a hypergraph.