Set Theory: 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 A ∪ B 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 X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

Solution:
X ∪ Y = {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 X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

Solution:
X ∪ Y = {1, 3, 5, 6, 8, 9}

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

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 ( X ∪ Y ) ’

b) Find ( X ∪ Y ) ’

Solution:
a) First, fill in the elements for X ∩ Y = {1}

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

Shade the region outside X ∪ Y to indicate (X ∪ Y ) ’

b) We can see from the Venn diagram that

(X ∪ Y ) ’ = {9}

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

(X ∪ Y ) ’ = {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) (A ∪ B ) ’ ii) (A ∪ C ) ’ iii) (A ∪ B ∪ C ) ’

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 A ∩ B ∩C = {3}, A ∩ B {1, 3},

A ∩ C = {3}, B ∩ C = {3} and then the other elements.

b) We can see from the Venn diagram that

i) (A ∪ B ) ’ = {10}

ii) (A ∪ C ) ’ = {2, 4, 6, 8}

iii) (A ∪ B ∪ C ) ’ = { }

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