Section 2.4: Problem 7 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
(a) Show that
.
(b) Show that
.
(a) Working backward (each line is sufficient to show the previous one). So, we need to show
or, equivalently,
.
- is inconsistent, by RAA.
-
Two things to show, by the definition of inconsistency:
- ,
- . We will show this one directly.
-
One thing to show:
- , by MP.
-
One thing to show:
- , by A3 and MP.
-
One thing to show:
- , by the Generalization Theorem. But this is just a tautology.
To show 2(b), we apply MP-rule twice to
and the tautology
.
(b) We need to show
. Working backward (each line is sufficient to show the previous one).
- , by EAV.
- , by the Generalization Theorem. We show that this holds directly.
By axiom group 2,
and
. By MP
,
.