# Section 1.2: Problem 3 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) Determine whether or not
is a tautology.

(b) Determine whether or not
tautologically implies
.

(a) The expression can be false only if both terms are false, but this would imply, on the one hand, that
and
, and, on the other hand, that
and
. Hence, this is a tautology.

(b) The first expression is false iff
is true and
is false, the second is false iff both terms are false iff
is true and
is true and
is false. Therefore, not only the first formula tautologically implies the second one, but also they are tautologically equivalent. Another way to see this:
is tautologically equivalent to
is tautologically equivalent to
is tautologically equivalent to
is tautologically equivalent to
.