Delta sets of numerical monoids

In this study, first of all the main concepts related to numerical semigroups, the irreducible numerical semigroups, symmetric and pseudo-symmetric numerical semigroups are mentioned. After that, the formulas and the connections representing the Delta set, Betti numbers, catenary degree, graphs and minimal presentation of the private pseudo-symmetric numerical semigroup family in the form of S =<3,3+ s,3+ 2s > with the embedding dimension and multiplicity three were obtained.