Set operations

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SET OPERATIONS



We know about set, now we will learn the set of operations. We can visualize relationship between set and set operations using VENN DIAGRAMS. 
  • COMPLEMENT OF A SET
  • UNION OF TWO SETS
  • INTERSECTION OF TWO SETS
  • DIFFERENCE OF TWO SETS
  • SYMMETRIC DIFFERENCE OF SETS
COMPLEMENT OF A SET   

The Complement of a set A is the set of all elements of U (the universal set) that are not in A. It is denoted by A′.

Venn diagram for complement of a set


For example,
   If  U = { a, e, i, o, u } and A = { i, u } then A′ = { a, e, o }

UNION OF TWO SETS

The union of two sets A and B is the set of all elements which are either in A or in B or in both. It is denoted by A∪B and read as A union B. 

Venn diagram for union of two sets


For example,
 If X = { red, yellow, white, blue } and Y = { black, green, pink, orange, brown }
 then X ⋃ Y = { red, yellow, white, blue, black, green, pink, orange, brown }

INTERSECTION OF TWO SETS

The intersection of two sets A and B is the set of all elements common to both A and B. It is denoted by A∩B and read as A intersection B.

Venn diagram for intersection of two sets


For example,
If A = { 2, 4, 6, 8, 10 } and B = { 1, 2, 3, 4, 5 } then A ⋂ B = { 2, 4 } since 2 and 4 are the common element of a sets A and B.

DIFFERENCE OF TWO SETS

Let A and B be two sets, the difference of sets A and B is the set of all elements which are in A, but not in B. It is denoted by A–B or A\B and read as A difference B. 

Venn diagram for difference of two sets


For example,
If A = { 1, 2, 3, 4, 5 } and B = { 4, 5, 6, 7, 8 } 
then A - B = { 1, 2, 3, 4, 5 } - { 4, 5, 6, 7, 8 }  = { 1, 2, 3 } 
       B - A =  { 4, 5, 6, 7, 8 } -   { 1, 2, 3, 4, 5 } = { 6, 7, 8 }


SYMMETRIC DIFFERENCE OF SETS

The symmetric difference of two sets A and B is the set (A–B)∪(B–A). It is denoted by AΔB.

Venn diagram for symmetric difference of sets








AΔB= (A–B) ∪ (B–A)






For example, 
If A = { 2, 4, 6, 8, 10 } and B = { 1, 2, 3, 4, 5 } then find A Δ B
SOLUTION:
A - B = { 6, 8, 10 }
B - A = { 1, 3, 5 }
AΔB= (A–B) ∪ (B–A)
AΔB= { 1, 3, 5, 6, 8, 10 }

FACTS:

  • John Venn was an English mathematician. He invented Venn diagrams which pictorially represent the relations between sets. 

  • Venn diagrams are used in the field of Set Theory, Probability, Statistics, Logic and Computer Science.