I have a question about the use of formal language. Could someone please explain me why this statement is logically equivalent?
f ∨ g ∧ h ≡ (f ∨ g) ∧ (f ∨ h).
I made two truth tables (see the pictures below), but this statement doesn´t hold (it isn't logically equivalent). However, according to the exercise it is. I hope someone can help me out (maybe I made a mistake).
My truth tables:
f ∨ g ∧ h:
(f ∨ g) ∧ (f ∨ h):
Thanks in advance.
Have a close look at the way the equivalence is denoted:
no parenthesis on the left side, two on the right specifying the disjunctions to be evaluated first.
If this was the conventional logical operator precedence, both pairs of parentheses were dispensable.
If left to right evaluation was conventional, the first pair was redundant: the convention actually is conjunction before disjunction (and the equivalence is denoted correctly as well as irredundant).
So, the f ∨ g column in your first table is immaterial, if correct.
The f ∨ g ∧ h column is inconsistent: the label with explicit parentheses would need to be f ∨ (g ∧ h), the values tabulated are for (f ∨ g) ∧ h.
