Linguistics Encyclopedia.pdf

Two- and three-membered sequences are referred to as ordered pairs and ordered
triples respectively. ‘’ are employed when describing sequences.
Sequences can be defined in terms of the more fundamental notions of set theory. This
can be done in several ways. For example, if we replace each member of the sequence by
a set containing that member together with its predecessors, we ensure that the nth
member of the sequence is distinguished by itself being an n-membered set. Thus, our
two sequences above could be expressed set-theoretically as follows:
(3) {{1}, {{1} 2}, {{1} {{1} 2} 3}}
(4) {{3}, {{3} 1}, {{3} {{3} 1} 2}}
(3) is a three-membered set, all three members themselves being sets. The number three
occurs as a member of just one of the members of (3), and this member is itself a threemembered
set. The only member of (4) of which the number three is a member is a onemembered
set. Hence, its different positions in the sequences (1) and (2) are reflected in
the set-theoretic presentation.
Sequences are useful in studying the properties of functions and relations. Some
examples will introduce the notion of a function:
4=the square of 2
81=the square of 9
London=the capital of the United Kingdom
Paris=the capital of France.
The squaring function determines a unique number as value when applied to any number
as argument: the capital function determines a unique city as value when applied to a
nation as an argument, and so on. The square function yields a value when applied again
to the results of an earlier application; the capital function is undefined for arguments
which are not nations. Hence, while
16=the square of the square of 2
there is not a value assigned by the capital function to
the capital of the capital of France,
or, of course, to
the capital of Paris
We express that a is a function of b thus:
a=f(b)
The linguistics encyclopedia 542
The arguments of a function are drawn from a set which is called its domain. The
domain of the squaring relation is the set of positive integers. The domain of the capital
function is the set of nations. The values of the function for its different arguments belong