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4.7.  Basic Set Operations

Definition 4.6 (Union)

The union of two sets A and B is denoted by: A [cup] B
and defined by the condition: for all x, x [isin] A [cup] B: x [isin] A or x [isin] B.

Equivalently, A [cup] B is given by: A [cup] B = { x | x [isin] A or x [isin] B }

Examples: numbers

[larr] Venn diagram

Definition 4.7 (Intersection)

The intersection of two sets A and B is denoted by: A [cap] B
and defined by the condition: for all x, x [isin] A [cap] B: x [isin] A and x [isin] B.

Equivalently, A [cap] B is given by: A [cap] B = { x | x [isin] A and x [isin] B }

Definition 4.8 (Disjoint)

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

Examples: numbers


[larr] Venn diagram

Examples: functions

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 [isin] A \ B: x [isin] A and x [notin] B.

Equivalently, A \ B is given by: A \ B = { x | x [isin] A and x [notin] B }

[larr] Venn diagram

Examples: numbers

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 [notin] A }

[larr] Venn diagram

Definition 4.11 (Powerset)

For any set S, the powerset of S is set Pow(S) defined by for all M, M [isin] Pow(S) if M [sube] S

Equivalently, Pow(S) = { M | M [sube] S }

Examples: numbers


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Created by unroff & hp-tools. © by Hans-Peter Bischof. All Rights Reserved (1998).

Last modified: 27/July/98 (12:14)