We show, step-by-step, that the set of all 2 x 2 matrices with real entries forms a ring under addition and multiplication.
We show that the set of multiples of any integer forms a subroup of the set of all integers.
We show that the set of all mapping of real numbers with the operations of sums and products forms a commutative ring with identity.
We show that the set of all mappings from a set to itself with composition as an operation is a monoid, but not a group.
We show that the set of symmetries of an equilateral triangles forms a group under composition.