Set Theory: Union Of Sets



In this lesson, we will learn the union of sets and the complement of the union of sets.

Related Topics: More Lessons on Sets.

Union of Sets

The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by AB and is read ‘A union B


Example :

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

Find XY and draw a Venn diagram to illustrate XY.

Solution:

XY = {1, 2, 3, 4, 5, 6, 7, 8} ←1 is written only once.





If X Y then X Y = Y. We will illustrate this relationship in the following example.

Example:

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}

X = {1, 6, 9} and Y = {1, 3, 5, 6, 8, 9}

Find XY and draw a Venn diagram to illustrate XY.

Solution:

X Y = {1, 3, 5, 6, 8, 9}



Complement of the Union of Sets

The complement of the set XY is the set of elements that are members of the universal set U but are not in XY. It is denoted by (XY ) ’

Example:

Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

a) Draw a Venn diagram to illustrate ( XY ) ’

b) Find ( XY ) ’

Solution:

a) First, fill in the elements for XY = {1}

Fill in the other elements for X and Y and for U

Shade the region outside XY to indicate (XY ) ’

b) We can see from the Venn diagram that

(XY ) ’ = {9}

Or we find that XY = {1, 2, 3, 4, 5, 6, 7, 8} and so

(XY ) ’ = {9}



Example:

Given U = {x : 1 ≤ x ≤10, x is an integer}, A = The set of odd numbers, B = The set of factors of 24 and C = {3, 10}.

a) Draw a Venn diagram to show the relationship.

b) Using the Venn diagram or otherwise, find:

i) (AB ) ’ ii) (AC ) ’ iii) (ABC ) ’

Solution:

A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 6, 8} and C = {3, 10}

a) First, fill in the elements for ABC = {3}, AB {1, 3},

AC = {3}, BC = {3} and then the other elements.

b) We can see from the Venn diagram that

i) (AB ) ’ = {10}

ii) (AC ) ’ = {2, 4, 6, 8}

iii) (AB C ) ’ = { }

Videos

Sets: Union and Intersection.



A mathematics lesson on set operation of union.







Union of Sets



The following video describes the Union and Intersection of Sets







Union, Intersection and Complement



Venn Diagrams: Shading Regions. This video shows how to shade the union, intersection and complement of two sets.







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