9.3. RELATIONS AND FUNCTIONS
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An equivalence relation creates a partition of the set A as a collection of
nonempty, pairwise disjoint sets whose union is A.
Such a partition can also be used to define an equivalence relation by using a
is related to b if they are in the same subset.
Example. The set A = {1, 2, 3, 4, 5, 6, 7, 8} with the relation “has more
factors than” is transitive, but not reflexive or symmetric. The relation “is not
equal to” on the same set is symmetic, but not reflexive or transitive.
Definition. A function is a relation that matches each element of a first
set to an element of a second set in such a way that no element in the first set
is assigned to two di↵erent elements in the second set, i.e., is a relation where
no two ordered pairs have the same first element.
A function f that assigns an element of a set A to an element of set B is written
f : A ! B. If a 2 A, the function notation for the element in B assigned to
a is f (a), i.e., (a, f (a)) is an ordered pair of the function (also relation) f . A
is the domain of f and B the codomain of f . The set {f (a) : a 2 A} is the
range of f . The range is a subset of the codomain.
Example.
1) None of our previous examples of relations were functions.
2) A sequence, a list of numbers arranged in order, called terms, is a function
whose domain is the set of whole numbers. “1” is matched with the first or
initial term, “2” with the second term, etc.