A note on Sets and its Applications: CAT Quantitative Aptitude

Sets

Sets are well-defined collection of objects.

The order in which the elements of a set are listed does not matter.

If A is a subset of B, we sometimes say that A is ‘contained’ in B.
if and only if there exists an element x such that .

Operations on Sets

Two or more sets can be combined to produce new sets by using ‘operations on sets’.

These operations are:

a. Union of Sets

If A and B are two sets, then the union of A and B is the set which contains all those objects which are members of A or of B (or of both).

We denote this set by

Thus,

b. Intersection of Sets

The intersection of two sets A and B, denoted by , is the set of all objects that belong to both A and B. Thus,

c. Difference of Sets

For A and B, we get two difference sets denoted by A – B and B - A (or by A \ B, B \ A) given by

d. Complement of a Set
The complement of a set A, denoted by A^c or A’ is the set of all those objects which are not members of A. Thus,

e. Symmetric Difference

The symmetric difference of two sets A and B, denoted by A Δ B, is the union of A – B and B – A. Thus,

Venn Diagrams

Venn diagrams are pictorial description of sets and are very useful in understanding sets and their relationships. A big rectangle is drawn denoting the universal set U. All subsets of U are denoted by closed figures lying within this rectangle. The area or space (or dots in it) inside a closed figure denotes the members of the set.

Counting the Number of Elements

For this, we will concentrate on finite sets. n (A), #A, |A| are all used to denote the number of elements in A. We will use n(A).

The following results are useful (from now on, n (U) = N).

Some other useful results may also be derived from the above results. These will be taken up under application based problems on Sets.