Set Operations
The symbols U,∩, ‘ and others are called set operations. They are the
same as +, -,X and ÷ in Operations on real numbers. Special care must
be taken when dealing with them. Luckily, there is a simple rule like
BODMAS for set operation in a case like below;
AUB ∩ C’ U (BUD’)
It is advisable to start with Brackets, then complements. The others do
not matter which you start with as either U or ∩.
Let’s solve some examples below;
Note: The universal set E contains all elements under discussion and
therefore should be used especially when getting complements.
Examples
Given E= {3,4,5,6,7}, A={3,4,5}, B={5,6}, C=6
List
a) AUB
b) A∩B
c) B’
d) AUB’
e) A’
f) A’UB’
g) (AUB)’
h) (A’UB’)’
i) AUB∩C
j) AUB∩C’
k) (AUB)U(A∩C)’
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Solution
a) AUB
Step 1: Replace the set names with actual sets
AUB= {3,4,5}U{5,6}
Note: The set operation U has not being changed.
Step 2: Apply the operation, in this case join the sets.
AUB= {3,4,5,6}
b) A∩B
Doing the same two steps
A∩B= {3,4,5}∩{5,6}
A∩B={5}
c) B’
B’= {5,6}’ Here, the ‘ symbol needs us to write elements in E that
are not in B(members other than 5 and 6)
B’= { 3,4,7}
d) AUB’
AUB’={3,4,5}U{5,6}’, Next, apply the complement on B i.e on {5,6}
AUB’={3,4,5}U{ 3,4,7}
AUB’={3,4,5,7}
e) A’
A’={3,4,5}’
A’={6,7}
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f) A’UB’
A’UB’={3,4,5}’U{5,6}’
A’UB’={6,7}U{3,4,7}
A’UB’={3,4,6,7}
g) (AUB)’
(AUB)’=({3,4,5}U{5,6})’
solved first
sets inside the brackets must be
(AUB)’=({3,4,5,6})’ We now have a single set inside the brackets
and we can remove them
(AUB)’={3,4,5,6}’
Apply complement
(AUB)’={7}
h) (A’UB’)’
(A’UB’)’=({3,4,5}’U{5,6}’)’ Brackets first
(A’UB’)’=({6,7}U{3,4,7})’
(A’UB’)’=({3,4,6,7})’
(A’UB’)’={3,4,6,7}’
(A’UB’)’={5}
i) AUB∩C
AUB∩C= {3,4,5}U{5,6}∩ {6} Here you can start with either U or ∩
, 𝑙𝑒𝑡𝑠 𝑠𝑡𝑎𝑟𝑡 𝑤𝑖𝑡ℎ 𝑈
AUB∩C= {3,4,5,6} ∩ {6}
AUB∩C= {6}
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j) AUB∩C’
AUB∩C’= {3,4,5}U{5,6}∩ {6}′
Deal with complement first
AUB∩C’= {3,4,5}U{5,6}∩ {3,4,5,7}
AUB∩C’= {3,4,5,6}∩ {3,4,5,7}
AUB∩C’= {3,4,5}
K) (AUB)U(A∩C)’
(AUB)U(A∩C)’= ({3,4,5}U{5,6})U({3,4,5}∩ {6})’ Brackets first
(AUB)U(A∩C)’= {3,4,5,6}U{}’ Complement second
(AUB)U(A∩C)’= {3,4,5,6}U{3,4,5,6,7}
(AUB)U(A∩C)’= {3,4,5,6,7}
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