Question 6: In this exercise, we will denote Boolean variables by lowercase letters, such as p and q. A proposition is any Boolean formula that can be obtained by applying the following recursive rules:

1. For every Boolean variable p, p is a proposition.

2. If f is a proposition, then f is also a proposition.

3. If f and g are propositions, then (f g) is also a proposition.

4. If f and g are propositions, then (f g) is also a proposition

Let p and q be Boolean variables. Prove that is a proposition Let denote the not-and operator. In other words, if f and g are Boolean formulas, then (f tg) is the Boolean formula that has the following truth table (0 stands for false, and 1 stands for true); 0 Let p be a Boolean variable. Use a truth table to prove that the Boolean formulas (pt p) and-p are equivalent Let p and q be Boolean variables. Use a truth table to prove that the Boolean formulas ((pt p) (q q)) and p V q are equivalent Let p and q be Boolean variables. Express the Boolean formula (p q) as an equivalent Boolean formula that only uses the t-operator. Use a truth table to justify your answer. Prove that any proposition can be written as an equivalent Boolean formula that only uses the ft-operatoir