4.7.  Basic Set Operations

Definition 4.6 (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 4.7 (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 4.8 (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 * , -1 * ), (+1 * , +1 * ) }

Definition 4.9 (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 4.10 (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 4.11 (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: 27/July/98 (12:14)