Discrete
Math
Section 2.2: Set Operations
Dr. Jason Snyder
Collin College
Definition 1
Let 𝐴 and 𝐵 be sets. The union of the sets 𝐴 and 𝐵, denoted by 𝐴 ∪ 𝐵, is the set that
contains those elements that are either in 𝐴 or in 𝐵, or in both.
That is 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵}.
Example 1
Let 𝐴 = {1,2,3,5} and let 𝐵 = {3,4,5,6}. List the elements of 𝐴 ∪ 𝐵 using the roster
method.
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Definition 2
Let 𝐴 and 𝐵 be sets. The intersection of the sets 𝐴 and 𝐵, denoted by 𝐴 ∩ 𝐵, is the
set that contains those elements in both 𝐴 and 𝐵.
That is 𝐴 ∩ 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵}.
Example 2
Let 𝐴 = {1,2,3,5} and let 𝐵 = {3,4,5,6}. List the elements of 𝐴 ∩ 𝐵 using the roster
method.
Definition 3
Two sets are called disjoint if their intersection is the empty set.
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Definition 4
Let 𝐴 and 𝐵 be sets. The difference of 𝐴 and 𝐵, denoted by 𝐴 − 𝐵, is the set
containing those elements that are in 𝐴 but not in 𝐵. The difference of 𝐴 and 𝐵 is also
sometimes called the complement of 𝐵 in 𝐴.
That is 𝐴 − 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵}.
The difference of set 𝐴 and 𝐵 is sometimes denoted by 𝐴 ∖ 𝐵.
Example 3
Let 𝐴 = {1,2,3,5} and let 𝐵 = {3,4,5,6}. List the elements of 𝐴 − 𝐵 and 𝐵 − 𝐴 using the
roster method.
Definition 5
Let 𝑈 be a universal set. The complement of the set 𝐴, denoted by 𝐴, is the difference
of 𝑈 and 𝐴. That is 𝐴 = 𝑈 − 𝐴.
Example 4
For this problem, let ℤ be the universal set. Let 𝐴 = {𝑥 ∈ ℤ|𝑥 ≥ 10}. What is the set 𝐴?
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Set Identities
Identity Laws
 𝐴∩𝑈 =𝐴
 𝐴∪∅=𝐴
Domination Laws
 𝐴∪𝑈 =𝑈
 𝐴∩∅=∅
Idempotent Laws
 𝐴∪𝐴 =𝐴
 𝐴∩𝐴 =𝐴
Complementation Law
(𝐴̅) = 𝐴
 ̅̅̅̅̅
Commutative Laws
 𝐴∪𝐵 =𝐵∪𝐴
 𝐴∩𝐵 =𝐵∩𝐴
Associative Laws
 𝐴 ∪ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∪ 𝐶
 𝐴 ∩ (𝐵 ∩ 𝐶) = (𝐴 ∩ 𝐵) ∩ 𝐶
Distributive Laws
 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)
De Morgan’s Laws
 ̅̅̅̅̅̅̅
𝐴 ∪ 𝐵 = 𝐴̅ ∩ 𝐵̅
 ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵 = 𝐴̅ ∪ 𝐵̅
Absorption Laws
 𝐴 ∪ (𝐴 ∩ 𝐵) = 𝐴
 𝐴 ∩ (𝐴 ∪ 𝐵) = 𝐴
Complement Laws
 𝐴 ∪ 𝐴̅ = 𝑈
 𝐴 ∩ 𝐴̅ = ∅
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Example 5
Prove that ̅̅̅̅̅̅̅
𝐴 ∪ 𝐵 = 𝐴̅ ∩ 𝐵̅.
Example 6
Prove that 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
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Example 7
Prove that ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐶̅ ∪ 𝐵̅) ∩ 𝐴̅ .
Example 8
Let 𝐴 = {0,2,4,6,8}, 𝐵 = {0,1,2,3,4}, and 𝐶 = {0,3,6,9}. What are 𝐴 ∪ 𝐵 ∪ 𝐶 and
𝐴 ∩ 𝐵 ∩ 𝐶?
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Definition 6
Let 𝐴1 , 𝐴2 , …, 𝐴𝑛 be a collection of 𝑛 sets. The union of these sets, 𝐴1 ∪ 𝐴2 ∪ ⋯ ∪ 𝐴𝑛 is
the set containing those elements that belong to at least of the sets 𝐴1 , 𝐴2 , …, 𝐴𝑛 .
This union is denoted by
𝑛
𝐴1 ∪ 𝐴2 ∪ ⋯ ∪ 𝐴𝑛 = ⋃ 𝐴𝑖 .
𝑖=1
Definition 7
Let 𝐴1 , 𝐴2 , …, 𝐴𝑛 be a collection of 𝑛 sets. The intersection of these sets,
𝐴1 ∩ 𝐴2 ∩ ⋯ ∩ 𝐴𝑛 is the set containing those elements that belong to all of the sets 𝐴1 ,
𝐴2 , …, 𝐴𝑛 .
This union is denoted by
𝑛
𝐴1 ∩ 𝐴2 ∩ ⋯ ∩ 𝐴𝑛 = ⋂ 𝐴𝑖 .
𝑖=1
Example 9
Let 𝐴𝑖 = {𝑖, 𝑖 + 1, 𝑖 + 2, … } for 𝑖 = 1, 2, 3, 4, …
Find
𝑛
⋃ 𝐴𝑖
𝑖=1
𝑛
and
⋂ 𝐴𝑖
𝑖=1
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This can be extended to other, even infinite, collections of sets.
Example 10
Let 𝐴𝑖 = {𝑖, 𝑖 + 1, 𝑖 + 2, … } and 𝐵𝑖 = {1,2,3, … , 𝑖} for 𝑖 = 1, 2, 3, 4, …
Find
∞
⋃ 𝐴𝑖
∞
and
⋂ 𝐴𝑖 .
𝑖=1
𝑖=1
∞
∞
Also find
⋃ 𝐵𝑖
𝑖=1
and
⋂ 𝐵𝑖 .
𝑖=1
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