4.7.  Basic Set Operations

Definition 1.6 (Union)

The union of two sets A and B is denoted by: A ∪ B
and defined by the condition: for all x, x ∈ A ∪ B: x ∈ A or x ∈ B.

Equivalently, A ∪ B is given by: A ∪ B = { x | x ∈ A or x ∈ B }

Examples: numbers

• A = { 3, 4 }; B = { 5 }. Then
  A ∪ B = { 3, 4, 5 }
• A = { 3, 4, 5 }; B = { 5 }. Then
  A ∪ B = { 3, 4, 5 }
• Let A = { x ∈ Nat | 4 < x ≤ 6 }, B = { x ∈ Nat | 7 ≤ x ≤ 8 }. Then A ∪ B = { x ∈ Nat | 4 < x ≤ 8 }

→ Venn diagram

Definition 1.7 (Intersection)

The intersection of two sets A and B is denoted by: A ∩ B
and defined by the condition: for all x, x ∈ A ∩ B: x ∈ A and x ∈ B.

Equivalently, A ∩ B is given by: A ∩ B = { x | x ∈ A and x ∈ B }

Definition 1.8 (Disjoint)

Two sets that have no common members are called disjoint sets.

Examples: numbers

• A = { 3, 4, 5 }; B = { 5 }. Then
  A ∩ B = { 5 }
• A = { 3, 4 }; B = { 5 }. Then
  A ∩ B = Ø
• Let A = { x ∈ Nat | 4 < x ≤ 8 }, B = { x ∈ Nat | 7 < x ≤ 9 }.
  Then A ∩ B = { x ∈ Nat | 7 < x ≤ 8 }
• Let A = { x ∈ Nat | 4 < x ≤ 6 }, B = { x ∈ Nat | 7 < x ≤ 9 }.
  Then A ∩ B = Ø (disjoint)

→ Venn diagram

Examples: functions

• Let A = { (x, y) | y = x + 3 }, B = { (x, y) | y = -x + 1 }.
  Then A ∩ B = { (x, y) | (-1, 2) }
• Let A = { (x, y) | y * y + x * x = 1 }, B = { (x, y) | y = x }.
  Then A ∩ B = { (x, y) | (-1 * sqrt(1/2), -1 * sqrt(1/2) ), (sqrt(1/2), sqrt(1/2)) }

Definition 1.9 (Relative Complement)

The relative complement of two sets A and B is denoted by: A \ B
and defined by the condition: for all x, x ∈ A \ B: x ∈ A and x ∉ B.

Equivalently, A \ B is given by: A \ B = { x | x ∈ A and x ∉ B }

→ Venn diagram

Examples: numbers

• A = { 3, 4, 5 }; B = { 5 }. Then
  A \ B = { 3, 4 }
• A = { 3, 4 }; B = { 5 }. Then
  A \ B = { 3, 4 }
• Let A = { x ∈ Nat | 4 < x ≤ 8 }, B = { x ∈ Nat | 7 ≤ x ≤ 9 }. Then
  A \ B = { x ∈ Nat | 4 < x < 7 }
  A \ B = { x ∈ Nat | 4 < x ≤ 6 }
• Let A = { x ∈ Nat | 4 < x ≤ 6 }, B = { x ∈ Nat | 7 < x ≤ 9 }. Then A ∩ B = Ø

Definition 1.10 (Complement)

Let A be a set. The complement of A, written ~A, is set set of all objects not belonging to A.

We write: ~A = { x | x ∉ A }

→ Venn diagram

Definition 1.11 (Powerset)

For any set S, the powerset of S is set Pow(S) defined by for all M, M ∈ Pow(S) if M ⊆ S

Equivalently, Pow(S) = { M | M ⊆ S }

Examples: numbers

• Let S1 = { a, b }.
  Then the subsets of S1 are Ø, { a }, { b } , { a, b } → Pow(S) = { Ø, { a }, { b } , { a, b } }

Last modified 22/May/97