Universal Set

In this lesson, we will learn what is a universal set and representation of universal set in venn diagrams.

Definition:

The set that contain all the elements of a given collection, and therefore all sets are subset of the Universal Set is known as the Universal Set.

Symbol: Universal Set is represented by symbol “U”.

Example: let  A = {1, 2, 3}, B = { 2, 3, 4, 5} and 

C = {5, 6, 7, 8}

Universal Set have all the elements of set A, B and C

∴ U = {1, 2, 3, 4, 5, 6, 7, 8}

Note: All the sets are subsets of Universal Set.

         Therefore,     A ⊂ U

                             B ⊂ U

                             C ⊂ U

Presentation of Universal Sets in venn diagrams,

Usually relationship between sets represented by diagrams which are known as venn diagrams.

In venn diagrams the Universal Set is represented by a rectangle and it’s labelled  by U, and other elements(subsets) of the given set is written in their respective circles.

Step 1: First draw a rectangle, and label it by U, to represent the universal set.

Step 2: Draw circles within the rectangle to represent the subsets of the universal set.

Step 3: Write the remaining elements outside the circles but within the rectangle.

Example 1: Draw a venn diagram to represent the sets?

U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 3}, B = {4, 5}

In above example A and B have no common elements, this relationship is shown in venn diagram below. 

Example 2: Draw a venn diagram to represent the sets?

U = {Natural Numbers less than 10}

A = {Multiples of 2 less than 10}, 

B = {Odd Numbers less than 8} 

∴ U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

   A = {2, 4, 6, 8}, 

   B = {1, 3, 5, 7}

Example 3: Draw a venn diagram to represent the sets?

U = {a, b, c, d, e, f, g, h, i, j}

A = {a, b, c, d}, B = {c, d, e, f} and  C = {f, g, h}

In above example A and B have common elements, and B and  C have also common elements this relationship is shown in venn diagram below. 

Example 4: If  U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, is the universal set of which

  A = {2, 4, 6, 8, 10} and  B = {4, 6} are subsets, and also B⊂A.

Draw a venn diagram to represent the sets?