SUBSETS –
Sets whose all elements are contained in another set Y are called subsets of the set Y. For example if A = {1, 3, 5, 7, 9, 11} and B = {3, 9}, the B is the subset of A.
Definition – If A & B are sets such that every member of set A is a member of set B, then set A is called a subset of set B.
Since, by definition, if A = {x, y, z}, and B = {x, y, z} then A is a subset of B. also since A is equal to B, we use the symbol ⊆ for “is a subset of”. Thus, A ⊆ B, the symbol ⊆ means contained in or possibly equal to.
Proper Subset –
If a set Q = {a, c} is a subset of set P = {a, b, c, d} and set Q is not equal to set P, then set Q is called a proper subset of set P and is expressed as Q ⊂ P. The set P is called the superset of set Q. It is expressed as P ⊃ Q.
The symbol ⊊ indicates is not a subset of and the symbol ⊄ indicates is not a proper subset of.
⊆ - is a subset of
⊂ - is a proper subset of
⊄ - is not a proper subset of
⊃ - is a super-set of
Some Theorem On Subsets
Your second block of text...