The study of Sets, which are collections of distinct objects.
Fundamentals
Transclude of Set
Subset
A part of a Set; Set A is a subset of B.
Definition
A set
is a subset of if all elements of A are also elements of B. Proper Subset
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Superset
A Set encompassing/extending another set; Set B is a subset of A.
- A superset is the converse of a Subset. They are defined the same way, but the operands are swapped.
- Exclusively using subsets is favored
Definition
Proper Superset
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Set Builder Notation
A set with some Object such that
etc are true. e.g. Link to original
Universe
The Universe, Universal Set, or Universe of Discourse, denoted by
is the Set that contains all the entities one wishes to consider in a given situation. Link to original
- denoted mainly by
but also sometimes by
Venn Diagram
In Set Theory, a Venn Diagram is a diagram that visually demonstrates the relationship between Sets, their overlap, and their relation to the Universe of Discourse
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Principle of Inclusion-Exclusion
For 2 sets, for example
For 3 sets, for example
Link to originalNote that we add
Numbers
Number
An element of a Division Ring, used for measuring, counting, quantifying, etc.
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Set Name Symbol Description Examples Natural Number Counting numbers Whole Number or Nonzero Integers Integer Natural numbers, their negatives, and zero. Rational Number Any number expressible as a fraction . Irrational Number Numbers that cannot be fractions (non-repeating decimals). Real Number All points on the continuous number line ( ). Imaginary Number Complex Number Quaternion Octonion Takes up too much space
Functions
Transclude of Function
Domain
The Set of all inputs to a Function
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Codomain
The Set of all outputs of a Function
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Injective
If there is at most one location in the Codomain for every location in the Domain
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Surjective
If there is a location in the Codomain for every location in the Domain
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Bijective
If there is exactly one location in the Codomain for every location in the Domain for the Transformation
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