Question

Consider the proposition:

Fish will thrive if and only if the water is deep and not polluted.

Translated in propositional logic as follows:

(deep∧¬polluted)↔thrive

Rewrite the proposition without→,↔and using only∨,∧and¬ as needed.

Answer 1

Let there be any statement of the form for propositions P and Q. This statement says that P implies Q and Q implies P. Therefore an equivalent way of saying this is .

Now consider a statement of the form . If P is false, then nothing can be said about Q. But if P is true, then Q has to be true as well. Therefore either P is false or Q is true. Therefore an equivalent statement is .

The statement is . The can be replaced as
. Now replace each as:
. This is as required in the question.

The final answer is .

Comment in case of any doubts.