# Subset of a Set

A set A is said to be a subset of the set B is each element of the set A is also the element of the set B. The symbol used is '' i.e. A B <=> (x Îµ A => x Îµ B).

Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A B.

**e.g.** If A = {a, b, c, d} and B = {b, c, d} then B A or equivalently A B (i.e. A is a super set of B).

**Sets A and B are said to be equal if A B and B A and we write A = B.**

# Equality of Two Sets

# Universal Set

As the name implies, it is a set with collection of all the elements and is usually denoted by U. e.g. set of real numbers R is a universal set whereas a set A = [x : x < 3} is not a universal set as it does not contain the set of real numbers x > 3. Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to that set. e.g. If A = {x : x < 3 then (or A

^{c}) = complimentary set of A = {x : x > 3}. Hence we can say that A U = U i.e. Union of a set and its complimentary is always the Universal set and A âˆ© = f i.e. intersection of the set and its complimentary is always a void set. Some of the useful properties of operation on sets are as follows:

**Example:**

If A = {a, b, c} and B = {b, c, d} then evaluate A Ï… B, A âˆ© B, A - B and B - A.

Solution:

A U B = {x : x Îµ A or x Îµ B} = {a, b, c, d}

A âˆ© B = {x : x Îµ A or x Îµ B} = {b, c}

A - B = {x : x Îµ A and x B} = {a}

B - A = {x : x Îµ B and x A} = {d}