what is difference between suset, proper subset , improper subset

We will define a "set" to be an unordered group of objects with no duplicates. Note that the objects in the sets can themselves be sets. If a set has a finite number of objects, we can describe the set by enumerating all of the objects in it. For example, the set containing the positive integers from 1 to 5 is

{1, 2, 3, 4, 5}.If on the other hand we wish to describe an infinite set, such as the set of even positive integers, we use what is called "set builder notation":{x : x > 0 and x / 2 has no remainder}This is read verbally as "the set of all x such that x is greater than 0 and x divided by 2 has a zero remainder" (where the colon ":" is read "such that").There are two special sets: the "empty set" and the "universal set". The empty set (or null set) is the set which contains no objects and is denoted {}, or by the symbolAs is always the case for standard notation which is not available on keyboards, we will sometimes denote the empty set by the numeral 0; when confusion might arise, we will use {} instead. The universal set is denoted by the capital letter U.Two sets are equivalent if they have exactly the same objects in them. For example,

{a, b, c, d} and {c, a, d, b}are equivalent, while{a, b, c, d} and {{a, b}, c, d}are not since the former set is a set of four objects, while the latter set is a set with only three objects, one of which itself is a set. It is important to note that two sets which do not have the same number of objects cannot be equivalent. Two sets are "disjoint" if they have no objects in common.Set membership is notated using the symbol ∈:

a ∈ {a, b, c}This is read "a is a member of the set {a, b, c}" or "a is an element of the set {a, b, c}".A "proper subset" of a set A is simply a set which contains some but not all of the objects in A. Proper subsets are denoted using the symbolFor example, the set {a, b} is a proper subset of the set {a, b, c}:An "improper subset" is a subset which can be equal to the original set; it is notated by the symbolwhich can be interpreted as "is a proper subset or is equal to".Note that the empty set is a member of the universal set; it is also a subset of the universal set. In fact, the empty set is a subset of every set.

Relationships between multiple sets are sometimes graphically described using Venn Diagrams. A Venn Diagram describing the relationship between three sets A, B and C always begins with the following picture: