Section 2.2: Subsets and Set Operations
A universal set, symbolized by 𝑈 , is the set of all potential elements under
consideration for a specific situation.
EX: {a, b, c, ... y, z}
EX: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
(*) A clever method for visualizing sets and their relationships called a Venn
diagram.
The Complement of a Set
′
The complement of a set 𝐴, symbolized 𝐴 , is the set of elements contained
in the universal set that are not in 𝐴.
′
EXAMPLE 1 Let 𝑈 = {v, w, x, y, z} and 𝐴 = {w, y, z}. Find 𝐴 and draw
a Venn diagram that illustrates these sets.
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Subsets
Subsets If every element of a set 𝐴 is also an element of a set 𝐵, then 𝐴 is
called a subset of 𝐵. The symbol ⊆ is used to designate a subset; in this
case, we write 𝐴 ⊆ 𝐵.
(*) Every set is a subset of itself.
(*) The empty set is a subset of every set.
EX: If we start with the set {x, y, z}, how many subsets we can form?:
EXAMPLE 2 Find all subsets of 𝐴 = {American Idol, Survivor}.
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If a set 𝐴 is a subset of a set 𝐵 and is not equal to 𝐵, then we call 𝐴 a
proper subset of 𝐵, and write 𝐴 ⊂ 𝐵.
EXAMPLE 3 Find all proper subsets of {x, y, z}.
The symbol ∕⊆ is used to indicate that the set is not a subset.
The symbol ∕⊂ is used to indicate that the set is not a proper subset.
EXAMPLE 4 State whether each statement is true or false.
(a) {1, 3, 5} ⊆ {1, 3, 5, 7}
(b) {a, b} ⊂ {a, b}
(c) {x∣x ∈ N and x > 10} ⊂ N
(d) {2, 10} ∕⊆ {2, 4, 6, 8, 10}
(e) {r, s, t} ∕⊂ {t, s, r}
(f ) {Lake Erie, Lake Huron} ⊂ The set of Great Lakes
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EXAMPLE 5 State whether each statement is true or false.
(a) ∅ ⊂ {5, 10, 15}
(b) {u, v, w, x} ⊆ {x, w, u}
(c) {0} ⊆ ∅
(d) ∅ ⊂ ∅
Intersection of Sets
The intersection of two sets 𝐴 and 𝐵, symbolized by 𝐴
all elements that are in set A and in set B.
EX: Find 𝐴
∩
∩
𝐵, is the set of
𝐵 when 𝐴 = {10, 12, 14, 15} and 𝐵 = {13, 14, 15, 16, 17}
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EXAMPLE 7 If 𝐴 = {5, 10, 15, 20, 25}, 𝐵 = {0, 10, 20, 30, 40}, and 𝐶 =
{30, 50, 70, 90}, find
(a) 𝐴
∩
𝐵
(b) 𝐵
∩
𝐶
(c) 𝐴
∩
𝐶
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When the intersection of two sets is the empty set, the sets are said to be
∩
disjoint (𝐴 𝐵 = ∅).
EX: Find A B when A = {10, 12, 14} and B = {11, 13, 15} and draw a
Venn diagram that illustrates these sets.
∩
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Union of Sets
The union of two sets 𝐴 and 𝐵, symbolized by 𝐴
elements that are in either set 𝐴 or set 𝐵 (or both).
∪
𝐵, is the set of all
EX: Find 𝐴 𝐵 if 𝐴 = {5, 10, 15, 20} and 𝐵 = {5, 20, 30, 45} and draw a
Venn diagram that illustrates these sets.
∪
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EXAMPLE 8 If 𝐴 = {0, 1, 2, 3, 4, 5}, 𝐵 = {2, 4, 6, 8, 10}, and 𝐶 = {1, 3, 5, 7},
find each.
(a) 𝐴
∪
𝐵
(b) 𝐴
∪
𝐶
(c) 𝐵
∪
𝐶
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EXAMPLE 9 Let 𝐴 = {l, m, n, o, p}, 𝐵 = {o, p, q, r}, and 𝐶 = {r, s, t, u}.
Find each.
(a) (𝐴
∪
∩
𝐵)
(b) 𝐴 (𝐵
(c) (𝐴
∩
∩
∪
𝐵)
𝐶
𝐶)
∪
𝐶
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EXAMPLE 10 If 𝑈 = {10, 20, 30, 40, 50, 60, 70, 80}, 𝐴 = {10, 30, 50, 70}, 𝐵 =
{40, 50, 60, 70}, and 𝐶{20, 40, 60}, find each.
′
(a) 𝐴
∩
(b) (𝐴
(c) 𝐵
′
𝐶
∩
∪
′
𝐵)
(𝐴
′
∩
∩
𝐶
′
𝐶)
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