# Section 2.2: Problem 28 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

For each of the following pairs of structures, show that they are not elementarily equivalent, by giving a sentence true in one and false in the other. (The language here contains
and a two-place function symbol
.)

(a)
and
, where
is the usual multiplication operation on the real numbers,
is the set of non-zero reals, and
is
restricted to the non-zero reals.

(b)
and
, where
is the set of positive integers, and
is usual addition operation restricted to
.

(c) Better yet, for each of the four structures of parts (a) and (b), give a sentence true in that structure and false in the other three.

(a), (b) and (c). The following table lists some sentences and whether each one is true in each structure.

Symbol | Formula |

Exists ( ) | |

Exists ( ) or ( ) such that every number has its opposite | |

Exists ( ) | |

Absence of ( ) or ( ) | |

Exists number which is not the result of for any numbers (in particular, ) |

Formula | ||||

+ | ||||

+ | ||||

+ | ||||

+ | ||||

+ |

By the way, why
. Suppose there is a model
of
. Then
iff
only if
only if
which is logically equivalent to
.