If '', then is a true proposition.

If '', then is a false proposition.

If '', then is a true proposition.

If
'', then I will not consider to
be a proposition (unless *lucky
number* has been defined.)

`` and '' is true if and only if both of are true.

`` or '' is true if and only if at least one of is true.

``not '' is true if and only if is false.

Observe that in mathematics, ``or'' is always assumed to be inclusive or: If ``'' and ``'' are both true, then `` or '' is true.

`` and '' is false.

`` or '' is true.

`` or '' is true.

``not(not )'' is true if and only if is true.

For each element of **Q** let be the
proposition

``
''. Thus
= ``
'', so is true, while
= ``
'', so is false.
Here I consider to be a rule which assigns to each element of **Q** a proposition

Thus the rule defined in the previous paragraph is a proposition
form over **Q**. Note that a proposition form is neither true nor false,
i.e. a proposition form is not a proposition.

``
'' is true if and only if (( are both true)
or ( are both false)).

Ordinarily one would not make a statement like

`` )''

even though this is a true proposition. One writes `` '' in an argument, only when the person reading the argument can be expected to see the equivalence of the two statements and .

If and are propositions,then

Thus if we know that (3.7) is true, then we can conclude that is true. The statement `` '' is sometimes read as `` if and only if ''.

Let be an arbitrary real number. Then

Thus the set of all numbers that satisfy equation (3.9) is {2,3}.

Hence for all propositions and

(3.12) |

(3.14) |

In proposition 3.16, is false, is true, and is true.

In proposition 3.17, is false, is false, and is true.

The usual way to prove is to assume that is true and show that then must be true. This is sufficient by our convention in (3.11).

If and are propositions, then
``
'' is also a proposition, and

We will not make much use of the idea of two propositions being equal.
Roughly, two propositions are equal if and only if they are word
for word the same. Thus ``'' and ``'' are not equal
propositions, although they are equivalent. The only time I will
use an ``'' sign between propositions is in definitions. For
example, I might define a proposition form over **N** by
saying

for all ``'',

or

for all
.

The definition we have given for ``implies'' is a matter of convention,
and there is a school of contemporary mathematicians
(called constructivists)
who define
to be true only if a ``constructive'' argument
can be given that the truth of follows from the truth of .
For the constructivists, some of the propositions of the sort we use
are neither true nor false, and some of the theorems we prove are
not provable (or disprovable). A very readable description of the
constructivist point of view can be found in the article *Schizophrenia in
Contemporary Mathematics*[10, pages 1-10].

a) Give examples of propositions such that `` '' and `` '' are both true, or else explain why no such examples exist.

b) Give examples of propositions such that `` '' and `` '' are both false, or explain why no such examples exist.

c) Give examples of propositions such that `` '' is true but `` '' is false, or explain why no such examples exist.

**a)**- .
**b)**- .
**c)**- .
**d)**- . (Here assume .)
**e)**- .
**f)**- .
**g)**- .
**h)**- .

Problem: Let be the set of all real numbers such that
. Describe the set of all elements such that

ARGUMENT A: Let be an arbitrary element of . Then

Hence the set of all real numbers that satisfy inequality (3.23) is the set of all real numbers such that .

ARGUMENT B: Let be an arbitrary element of . Then

Now

Hence the set of all real numbers that satisfy inequality (3.23) is the set of all such that either or .