# Section 2.2: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Assume that the language has equality and a two-place predicate symbol
. For each of the following conditions, find a sentence
such that the structure
is a model of
iff the condition is met.

(a)
has exactly two members.

(b)
is a function from
into
. (A

*function*is a single-valued relation, as in Chapter 0. For to be a function from into , the domain of must be all of ; the range of is a subset, not necessarily proper, of .)
(c)
is a permutation of
; i.e.,
is a one-to-one function with domain and range equal to
.

(a)
(there are at least two elements and among any three elements there is a pair of equal elements).

(b)
(
and
is single-valued).

(c)
(
and
and
is single-valued and
is injective).