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Set, Relations and Functions, . - Coggle Diagram
Set, Relations and Functions
Cardinality of a Set
Cardinality of the set M, |M| (sometimes also written as #M)
denotes the number of elements in M.
If |T|=0 then we also write T = (empty set)
If |S| N, then we say S is finite. Otherwise, we say S is infinite
Power Set
Power set of S, P(S) or 2S
, is the set of all subsets of the set S
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If S is finite, then |P(S)| = 2|S|
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elements of set A, ie. |P(A)| > |A|.
Relations
Generalizing the definition of ordered pairs we define to be an ordered n-tuple (or a sequence of length n)
We can explicitly name empty sequences, singlets, pairs, triples, quadruples,
quintuples, …, n-tuples.
Cartesian Product
Cartesian product of set A and set B, A × B, is the set of all possible
ordered pair (a, b) from A and B.
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Relations
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structure called a relation, which is just a subset of the Cartesian product
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Inverse Relations
If R = {(a, b)} then, the inverse relation of R, R−1 = {(b, a)}.
Composition of Relations
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The composite of R and S is the relation consisting (a, c), where a A, c C,
and for which there exixts an element b B such that (a, b) R and (b, c) S,
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S R = {(a, c) | bB, aRb bSc} is called the composite of R AB and S
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Equivalence Relations
A relation R on a set A is an equivalence relation if R is reflexive, symmetric,
and transitive
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Hasse Diagram
A Hasse diagram is similar to the digraph of R ignoring reflexive and
transitive relations and putting a graphically below b if aRb.
Functions
A function f from a nonempty set A to a nonempty set B is an assignment of exactly one element of B to each element of A
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Image of Subset
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The image of S, f(S), under the function f is the subset of B that consists of the
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Graph of a functions
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The graph of the function f is the set of ordered pairs {(a, b) | a A f(a) = b}. It
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