Types of Sets

1. Null set: A set with zero or no elements is called a Null set. It is denoted by { } or Ø. The null set cardinal number is 0.
2. Singleton set: Sets with only one element in them are called singleton sets.
   Ex.  {2}, {a}, {0}
3. Finite and Infinite set: A set having a finite number of elements is called a finite set. A set having infinite or uncountable elements in it is called an infinite set.
4. Universal set: A set which contains all the elements of all the sets and all the other sets in it, is called a universal set.
5. Subset: A set is said to be a subset of another set if all the elements contained in it are also part of another set.
   Ex. If A = {1,2}, B = {1,2,3,4} then, Set “A” is said to be subset of set B.
6. Equal sets: Two sets are said to be equal sets when they contain the same elements.
   Ex. A = {a,b,c} and B = {a,b,c} then A and B are called equal sets.
7. Disjoint sets: When two sets have no elements in common than the two sets are called disjoint sets.
   Ex. A = {1,2,3} and B = {6,8,9} then A and B are disjoint sets.
8. Power set: A power set is defined as the collection of all the subsets of a set and is denoted by P(A).

• If A = {a,b} then P(A) = { { }, {a}, {b}, {a,b} }
• For a set having n elements, the number of subsets is $$2^{n}$$

Undoubtedly one of the easiest parts of CAT. Most of the formulae in this section can be deduced logically with little effort. The difficult part of the problem is translating the sentences into areas of the venn diagram. While solving, pay careful attention to phrases like and, or, not, only, in as these generally signify the relationship.

Some other important properties

• A’ is called complement of set A, or A’ = U-A
  n(A-B) = n(A) - n(A∩B)
  A-B = A∩B’
  B-A = A’∩B
  (A-B) U B = A U B

Venn diagrams: A Venn diagram is a figure to represent various sets and their relationship.

I, II and III are the elements in only A, only B and only C respectively.

IV – Elements which are in all of A, B and C.
V - Elements which are in A and B but not in C.
VI – Elements which are in A and C but not in B.
VII – Elements which are in B and C but not in A.
VIII – Elements which are not in either A or B or C.

• Maxima and minima of overlap: As shown in the image, what is the maximum and minimum value of X?

• To maximize X, we must assume that for the smaller set A, A and B’=0. Hence all of A’s 64 elements are in A and B. Therefore max of X=64
• To minimize X, we must assume that the union of A and B is as large as possible. In this case, A or B=100. Hence, n(A or B)=n(A)+n(B)-n(A and B). Hence n(A and B)=64+82-100=46.

The similar concept applies for venn diagrams with three sets. Remember that,

To maximize overlap,

• Union should be as small as possible
• Calculate the surplus = n( A) +n(B) +n(C)-n(A or B or C)
• This can be attributed to $$ n(A\cap B\cap C')$$, $$n(A\cap B'\cap C)$$, $$n(A'\cap B\cap C)$$, $$ n(A\cap B\cap C)$$. To maximize the overlap, set the other three terms to zero.

To minimize overlap,

• Union should be as large as possible
• Calculate the surplus = n( A) +n(B) +n(C)-n(A or B or C)
• This can be attributed to $$n(A\cap B\cap C')$$, $$n(A\cap B'\cap C)$$, $$n(A'\cap B\cap C)$$, $$n(A\cap B\cap C)$$. To minimize the overlap, set the other three terms to maximum possible.