Problem With Logic - Substituting OR's and AND's and Negation
I am having trouble proving a logical problem (Exercise 1.26a from Elliott Mendelson's book 'Introduction to Logic'). The problem is as follows.
"If C is a statement involving only ~, AND and OR, and C` arises from C by replacing each AND with OR and each OR with AND, show that C is tautology if and only if ~C` is a tautology."
Here is my attempt:
First show only if.
Suppose there is a statement C which is A AND B (to take a simple case) and C is tautology.
then C` is equivalent to A OR B and C is always true
then ~C` is equivalent to ~(A OR B ) and A AND B is always true
then ~C` is equivalent to ~A AND ~B (~(A OR B ) is equivalent to ~A AND ~B) and A AND B is always true.
To me this does not lead to showing that ~C` is a tautology. Where have I gone wrong?