Combined Set Operations – Union, Intersection and Complement

Here, you'll learn how to perform combined set operations, i.e. union, intersection and set complement. Here are a few properties of mixed operators. This is true only if there are operators of the different types in the same problem.




The order of evaluation is not important.
A ∩ B = B ∩ A

Parentheses need to be evaluated first.
A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C)

Let's start with an example to contrast the difference between union and intersection, then move on to examples with combined operations.

Example 1 – Difference between union and intersect set operations
In a factory that consists of 120 skilled workers, 59 possess a diploma, 66 possess a degree, 20 possess neither a diploma nor a degree. Find
(a) the number of workers who possess both a diploma and a degree.
(b) the number of workers who possess a diploma or a degree.

Solution
Let U = {skilled workers in the factory}
Let D = {workers who possess a diploma}
Let I = {workers who possess a degree}

(a) n(D ∩ I) = x




Hence, the number of workers who possess both a diploma and a degree is 25 persons.

(b)


Hence, the number of workers who possess a diploma or a degree is 100 persons.

Example 2 – If set operations are performed in different order


The Venn diagram shows the relationship between three sets A, B, and C with the universal set, U. List the elements of
(a) (A ∩ B) ∪ C,
(b) A ∩ (B ∪ C),
(c) (A ∪ B) ∩ C,
(d) A ∪ (B ∩ C).

Solution
(a)
(A ∩ B) = {3, 5}
(A ∩ B) ∪ C = {3, 4, 5, 6, 9, 10}



(b)
(B ∪ C) = {3, 4, 5, 6, 7, 8, 9, 10}
A ∩ (B ∪ C) = {3, 4, 5}



(c)
(A ∪ B) = {1, 2, 3, 4, 5, 6, 7, 8}
(A ∪ B) ∩ C = {4, 5, 6}



(d)
(B ∩ C) = {5, 6}
A ∪ (B ∩ C) = {1, 2, 3, 4, 5, 6}




Example 3 – Additional examples of set operations
Based on the Venn diagram in Example 2, list the elements of
(a) A' ∩ (B ∪ C),
(b) (A' ∪ C') ∩ B.

Solution
(a)
(B ∪ C) = {3, 4, 5, 6, 7, 8, 9, 10}
A' = {6, 7, 8, 9, 10, 11, 12}
A' ∩ (B ∪ C) = {6, 7, 8, 9, 10}



(b)
A' = {6, 7, 8, 9, 10, 11, 12}
C' = {1, 2, 3, 7, 8, 11, 12}
(A' ∪ C') = {1, 2, 3, 6, 7, 8, 9, 10, 11, 12}
(A' ∪ C') ∩ B = {3, 6, 7, 8}




Example 4 - More combined set operations


The Venn diagram shows the universal set U and sets A, B, and C. On a separate Venn diagram, shade the region for
(a) (A ∪ B) ∩ C',
(b) (B ∩ C) ∪ (A ∪ B)',

Solution
(a)


(A ∪ B)



(A ∪ B) ∩ C'

(b)


(B ∩ C)



(B ∩ C) ∪ (A ∪ B)


Example 5


In the Venn diagram, A, B, and C are three sets whereas U is the universal set. The number of elements in a few subsets are shown and n(U) = 40. Find
(a) n(B)
(b) n[(A ∩ B) ∪ C],
(c) n[A' ∪ (B ∩ C')].

Solution
(a)


Create an equation from the information given by the Venn diagram.



(b)


n[(A ∩ B) ∪ C] = 8 + 14 = 22

(c)


n[A' ∪ (B ∩ C')] = 8 + 5 + 14 + 3 = 30



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i want proof for that by ven daigram Damien: Perhaps you could post a photo of what you've tried, and we can work on your solution.

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