Problem 6.13. A formula is in prenex form if it is written as a string of quantifiers (referred to as the prefix) followed by a quantifier-free part (referred to as the matrix).17 For each of the following formulas find a formula in the prenex form equivalent to the given formula and the matrix simplified as much as possible. (1) ¬(Vx)(Vy)(x + y = y + x) (2) ¬(Vx) (y)(x + y = 2013) (3) ¬(3x)(Vy) (2 > sin(y)) (4) ¬(Ve > 0)(38 > 0)(Vx)(Vz)(|xz|<d⇒ | sin(x)-sin(2)| < €). - - (5) ¬(Vx)(Ve > 0)(36 > 0)(V2)(x − z | <d⇒ |2 − 2 | < €). (6) (Vx)((x)=(3x)Þ(x)). (7) ¬(Vx) ((x) = (³x)Þ(x)).