Set Theory and Venn Diagrams

Sets

A set is well defined collection of distinct objects (also called as elements). Sets are conventionally denoted with capital letters. The elements of the sets are denoted by small letters. Sets A and B are equal if and only if they have precisely the same elements.

For example, The set of vowels of English Alphabet, A = {a, e, i, o, u}

Empty or Null Set

The set which contains no elements is called the empty set or the null set. The empty set is written as φ. Thus, φ = { }.

Singleton Set

A set containing only one element is called a singleton. For example, {1}, {4} are singleton sets.

Equality of Sets

The sets A and B are equal if they have same members that is if every elements of A is an element of B and every element of B is an element of A, then A = B

Finite and Infinite Set

The set which contains a definite number of elements is called a finite set. The set which contains an infinite number of elements is called an infinite set.

Disjoint Set

Two sets A and B are said to be disjoint, if they do not have any element in common. For example, A = {1, 2, 3} and B = {4, 5, 6} are disjoint sets.

Subset

If every element in set A is also an element of another set B. Then A is called a subset of B. Also, B is said to be super set of A.

A ⊆ B

The Null set φ is a subset of every set.

Proper Subset

A is a proper subset of B. if A ⊆ B and A ≠ B and is written as A ⊂ B. B contains at least one element more than A, then A is a proper subset of B.

Universal Set

If all the sets under consideration are the subsets of a fixed set U, then U is called the Universal set.

Operations on Sets

Union of Sets

Union of two sets A and B is the set of all elements which belongs to A or B (or to both) and is written as A ∪ B (A union B). For example,

If A = {1, 3, 5, 7, 9} and B = {2, 4, 5, 6, 9}, then A ∪ B = {1, 2, 3, 4, 5, 6, 7, 9}

Intersection of Sets

If A and B are any two sets, then intersection of A and B is the set of all elements which are in A and also in B. It is written as A ∩ B (A intersection B). For example, 

If A = {2, 4, 6, 8} and B = {4, 5, 6, 9}, then A ∩ B = {4, 6}

Difference of Sets

The difference of two sets A and B is set of elements which belongs to A but do not belong to B. This is written as A - B.

Demorgan Laws

If A, B, C are three sets, then

  • A - (B ∪ C) = (A - B) ∩ (A - C)
  • A - (B ∩ C) = (A - B) ∪ (A - C)

Complement of a Set

Let A be a subset of universal set U, then the complement of A is denoted by AC.

If U = {1, 2, 3, 4, 5, 6} and A = { 1, 3, 5}, then AC = {2, 4, 6}

Venn Diagrams

Venn diagrams are useful way of representing relationships between sets.

In a Venn diagram, the universal set is represented by a rectangle. Points inside the rectangle represent elements that are in the universal set; points outside represent things not in the universal set.

Other sets are represented by oval or circular loops, drawn inside the rectangle. Points inside a given loop represent elements in the set it represents; points outside represent things not in the set.

Disjoint Sets

The sets A and B are disjoint, because the loops don't overlap.

A ∩ B = φ

Subsets

A is a subset of B, because the loop representing set A is entirely enclosed by loop B.

A ∪ B = B

A ∩ B = A

A - B = φ

Two Sets Venn Diagram

A - B = ii

B - A = iv

A ∩ B = iii

Three Sets Venn Diagram