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Mathematics

Set Theory

Types of Sets

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Proper Subset

Definition: A set 'A' is a proper subset of a set 'B' if 'A' is a subset of 'B' but 'A' is not equal to 'B'. Example:

\{1, 2\} \subset \{1, 2, 3\}

Infinite

Definition: A set that does not have a countable number of elements. Example: The set of all natural numbers

\mathbb{N}

Overlapping Sets

Definition: Sets that have at least one element in common. Example:

\{1, 2\}

and

\{2, 3\}

overlap on the element '2'.

Disjoint Sets

Definition: Two sets are disjoint if they have no elements in common. Example:

\{1, 2\}

and

\{3, 4\}

are disjoint.

Intersection of Sets

Definition: The set of elements that are common to both 'A' and 'B'. Example: If

A = \{1, 2\}

and

B = \{2, 3\}

, then

A \cap B = \{2\}

Union of Sets

Definition: The set of all elements that are in 'A' or 'B' or both. Example: If

A = \{1, 2\}

and

B = \{2, 3\}

, then

A \cup B = \{1, 2, 3\}

Empty Set

Definition: A set with no elements. Example:

\emptyset

\{\}

Power Set

Definition: The set of all subsets of a set 'S', including the empty set and 'S' itself. Example: If

S = \{a, b\}

, then

P(S) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\}

Universal Set

Definition: A set that contains all the objects under consideration. Example: If considering natural numbers, the universal set is

\mathbb{N}

Singleton

Definition: A set with exactly one element. Example:

\{a\}

where 'a' is any element.

Subset

Definition: A set 'A' is a subset of a set 'B' if every element of 'A' is also an element of 'B'. Example:

\{1, 2\} \subseteq \{1, 2, 3\}

Finite

Definition: A set with a countable number of elements. Example:

\{1, 2, 3, 4, 5\}

Equal Sets

Definition: Two sets are equal if they contain exactly the same elements. Example:

\{a, b\}

and

\{b, a\}

are equal.

Complement of a Set

Definition: Consists of all elements that are in the universal set but not in 'A'. Example: If

U = \{1, 2, 3, 4\}

and

A = \{1, 2\}

, then the complement of 'A' is

\{3, 4\}

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