Study of algebraic structures using subsets was started by George Boole. After the invention of Boolean algebra, subsets were not used in any algebraic structures. In this book we develop algebraic structures using subsets of a set, or of a group, or semigroup, or a ring and we get algebraic structures. Using ring or semiring we get only subset semirings. BY the method we get an infinite number of non-commutative semirings of finite order. We build subset semivector spaces, and describe and develop interesting properties over them.