Exercise 2.38 (Essential Variables). Find the essential or non-essential variables of the following functions and find formulas without non-essential variables:

1 f(x1, x2, x3) = (x1 → (x1 ∨ x2)) → x3;

2 f(x1, x2) = ((x1 ∨ x2) → x2);

3 f(x1, x2, x3, x4) = (x1 ∨ x2 ∨ x2x3 ∨ x1x2x3)x4;

4 f(x1, x2) = (x1 ⊕ x2)(x1 ↓ x2);

5 f(x1, x2, x3) = (((x3 → x2) ∨ x1)(x2 → x1)x3x1) ⊕ x3;

6 f(x1, x2, x3) = (((x1 ∨ x2)(x1 ∨ x3) → x1) → x2x3)x2;

7 f(x1, x2, x3, x4) = (1011100111001010);

8 f(x1, x2, x3, x4) = (0011110011000011).