Page 10 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Definition 1: Equality of Sets
Any two sets are equal if that are identical. Thus the two sets S and T are equal, written as S = T
if they consist of exactly the same elements. When the two sets S and T are unequal, we write
S  T.
It follows from the definition that S = T if any one of x  S implies x  T and y  T implies y  S.
Also S is different from T (S +T) if there is at least one element in one of them which is not in the
other.

If every member of a given set S is also a member of T, then we say that S is a subset of T or
“S is contained in T” and write:

S  T
or equivalently we say that “T contains S” or T is a superset of S, and write
T  S
The relation

S  T
means that S is not a subset of T i.e. there is at least one element in T which is not in S.

Thus, you can easily see that any two sets S and T are equal if and only if S is a subset of T and T
is a subset of S i.e.
S = T  S  T and T  S.

If S  T but T  S, then we say that S is a proper subset of T. Note that S  S i.e. every set is a subset
of itself.

Another important concept is that of a set having no elements. Such a set, as you know, is called
an empty set or a null set or a void set and is denoted by O.

You can easily see that there is only one empty set i.e. O is unique. Further O is a subset of every
set.
Now why don’t you try an exercise?

Task Justify the following statements:
1. The set N is a proper subset of Z.
2. The set R is not a subset of Q.
3. If A, B, C are any three sets such that A  B, and B  C, then A  C.

So far, we have talked about the elements and subsets of a given set. Let us now recall the
method of constructing new sets from the given sets.

While studying subsets, we generally fix a set and consider the subsets of this set throughout our
discussion. This set is usually called the Universal set. This Universal set may vary from situations
to situations. For example, when we consider the subsets of R, then R is the Universal set.
When we consider the set of points in the Euclidean plane, then the set of all points in the
Euclidean plane is the Universal set. We shall denote the Universal set by X.

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