Let E and F be sets. With each element x of E, let there be associated a unique
element f(x) of F. Then f is called a function from E into F, and f is said to map E
into F. We write f : E 7! F to indicate it.
Let f be a function from E into F. For x in E, the point f(x) in F is called the
image of x or the value of f at x. Similarly, for A E, the set
{y 2 F : y = f(x) for some x 2 A}
is called the image of A. In particular, the image of E is called the range of f. Moving
in the opposite direction, for B F,
2.1 f?1(B) = {x 2 E : f(x) 2 B}
is called the inverse image of B under f. Obviously, the inverse of F is E.
Terms like mapping, operator, transformation are synonyms for the term “function”
with varying shades of meaning depending on the context and on the sets E and F. We
shall become familiar with them in time. Sometimes, we write x 7! f(x) to indicate
the mapping f; for instance, the mapping x 7! x3 + 5 from R into R is the function
f : R 7! R defined by f(x) = x3 + 5.
Injections, Surjections, Bijections
Let f be a function from E into F. It is called an injection, or is said to be injective, or
is said to be one-to-one, if distinct points have distinct images (that is, if x 6= y implies
f(x) 6= f(y)). It is called a surjection, or is said to be surjective, if its range is F,
in which case f is said to be from E onto F. It is called a bijection, or is said to be
bijective, if it is both injective and surjective.
These terms are relative to E and F. For examples, x 7! ex is an injection from R
into R, but is a bijection from R into (0,1). The function x 7! sin x from R into R is
neither injective nor surjective, but it is a surjection from R onto [?1, 1].
2. FUNCTIONS AND SEQUENCES 5
Sequences
A sequence is a function from N into some set. If f is a sequence, it is customary
to denote f(n) by something like xn and write (xn) or (x1, x2, . . .) for the sequence
(instead of f). Then, the xn are called the terms of the sequence. For instance,
(1, 3, 4, 7, 11, . . .) is a sequence whose first, second, etc. terms are x1 = 1, x2 = 3, ...
.
If A is a set and every term of the sequence (xn) belongs to A, then (xn) is said to
be a sequence in A or a sequence of elements of A, and we write (xn) A to indicate
this.
A sequence (xn) is said to be a subsequence of (yn) if there exist integers 1
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