The course is aimed at giving students additional practice and application of standard proof techniques, learnt earlier in the course in MA 211 Set Theory & Logic; and this course is also aimed at preparing students for higher level courses in pure mathematics. The algebraic theory of semigroups and algebraic theory of groups were chosen as the fields of study in which the students will apply their proof techniques. The Group Algorithm Programming (GAP) will be used to produce concrete examples of such abstract mathematical objects, as a way of producing examples and counterexamples (this software is freely available on the internet). Examples of such object will include: semigroups, monogenic semigroups, semigroup of full transformations, semigroup of binary relations, semilattices, strong semilattices of semigroups, symmetric groups, modular groups, Dihedral groups, and Alternating groups.