# Section 1.3: Problem 1 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

Rewrite the tautologies in the “selected list” at the end of Section 1.2, but using the conventions of the present section to minimize the number of parentheses.

Selected list of tautologies.

1. Associative and commutative laws for
,
,
:

for
,
,

for
,
.

2. Distributive laws:

.

.

3. Negation:

.

.

.

De Morgan’s laws:

.

.

4. Other

Excluded middle:
.

Contradiction:
.

Contraposition:
.

Exportation:
(we could have removed one more pair of parentheses on the right, but I do not particular like the fact that in this case the interpretation would be heavily dependent on our particular assumption that
is interpreted as
).