Proper subset definition

A proper subset of a setAAis asubsetofAAthat is not equal toAA. In other words, ifBBis a proper subset ofAA, then all elements ofBBare inAAbutAAcontains at least one element that is not inBB.

For example, ifA={1,3,5}A={1,3,5}thenB={1,5}B={1,5}is a proper subset ofAA. The setC={1,3,5}C={1,3,5}is a subset ofAA, but it is not a proper subset ofAAsinceC=AC=A. The setD={1,4}D={1,4}is not even a subset ofAA, since 4 is not an element ofAA.