Mathematics/Discrete Math/Sets

Sets, aka groups of elements and logic regarding such.

Terminology

• Set - A grouping of various elements.
• Subset - A grouping of such that all elements of the set are also contained within an equal-sized or larger set. May include two equivalent sets.
• Proper Subset - A subset in which the two sets are not equivalent.
• Empty Set/Zero Set (∅) - A set containing exactly 0 elements. Aka a null set.
• Closed Set - A set in which it's own boundary is contained within the set.
• Open Set - A set where the boundary is excluded from the set, but all values between the boundary are included.
• Cardinality - The number of distinct elements within a set. Ex: The set {-1, 0, 1} has a Cardinality of 3.

Common Set Types:

• Universal Set (U) - An arbitrary set, which changes based on context. However, it is assumed to always contain every possible element currently being considered for the given context.
• Integers (Z) - The set of all integers. Ex: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
• Positive Integers (Z⁺) - The set of all positive integers. Ex: {1, 2, 3, ...}.
• Natural Numbers (N) - The set of Z⁺, but including zero. Ex: {0, 1, 2, 3, ...}
• Positive Real Numbers (R+) - The set of non-imaginary numbers, greater than zero. Includes fractions, decimals, etc.
• Real Numbers (R) - The set of R+, but including zero and negative numbers.
• Complex Numbers (C) - The set of all numbers, including imaginary ones.