Set Theory- Basics & Applications for JKSSB FAA Posts
Tips, Formulae and Shortcuts for
Set Theory- Basics & Applications by Rayees Khan, Founder Team Kashmir Student
Tip 1:
▪ Set is defined as a collection of well-defined objects.
Ex. Set of whole numbers
▪ Every object is called Element of the set.
▪ The number of elements in the set is called cardinal number.
Tip 2:
Types of Sets
1. Null set:
A set with zero or no elements is called Null set. It is denoted by { } or Ø. 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 finite number of elements is called finite set. A set having infinite or
uncountable elements in it is called 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 universal set.
5. Subset:
A set is said to be 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 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 then 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 are 2^n (2 raised power n)
Tip 3:
Union of sets is defined as the collection of elements either in A or B or
both. It is represented by symbol “U”. Intersection of set is the collection of
elements which are in both A and B.
Tip 4:
Properties of Sets:
▪ The null set is a subset of all sets
▪ Every set is subset of itself
▪ A U (BUC) = (AUB) U C
▪ A ∩ (B∩C) = (A∩B) ∩ C
▪ A U (B∩C) = (AUB) ∩ (AUC)
▪ A ∩ (BUC) = (A∩B) U (A∩C)
▪ A U Ø = A
Tip 5:
▪ Let there are two sets A and B then,
n(AUB) = n(A) + n(B) - n(A∩B)
▪ If there are 3 sets A, B and C then,
n(AUBUC) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C)
Tip 6:
To maximize overlap,
▪ Union should be as small as possible
▪ Calculate the surplus = n(A) + n(B) + n(C) - n(AUBUC)
▪ This can be attributed to n(A∩B∩C′), n(A∩B′∩C), n(A′∩B∩C), n(A∩B∩C).
▪ To maximize the overlap, set the other three terms to zero.
Tip 7:
To minimize overlap,
▪ Union should be as large as possible
▪ Calculate the surplus = n(A) +n(B) +n(C) - n(AUBUC)
▪ This can be attributed to n(A∩B∩C′), n(A∩B′∩C), n(A′∩B∩C), n(A∩B∩C).
▪ To minimize the overlap, set the other three terms to maximum possible.
Tip 8:
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
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