Consider the set S = Z, and suppose that we define the addition + and multiplication . operations on this set in the usual way. Which of the following properties does this number system satisfy for all a, b, c E S? O (P1) a + (b + c) = (a + b) + c (associative law of addition) O (P2) a + 0 = a = 0 + a (existence of additive identity) O (P3) a + (-a) = 0 = (-a) + a (existence of additive inverses) O (P4) a + b = b + a (commutative law of addition) O (P5) a . (b . c) = (a . b) . c (associative law of multiplication) O (P6) a . 1 = a = 1 . a (existence of multiplicative identity) O (P7) a . al = 1 = al . a (existence of multiplicative inverses) O (P8) a . b = b . a (commutative law for multiplication) O (p9) a . (b + c) = a . b + a . c (distributive law) O (P10) either a a (trichotomy law) O (P11) a + b E S (closure under addition) O (P12) a . b E S (closure under multiplication) Consider each of the following real numbers, and select the smallest set (N, Z, Q, R) that each belongs to. (a) 3 (No answer given) * (b) 4 (No answer given) * (c) 1 (No answer given) * (d) - (No answer given) + (e) 8 (No answer given) (No answer given) * (g) -2 (No answer given) + (h) (No answer given) + (i) (No answer given) + (j) -12 (No answer given) * Consider the set S = 8N + 3 = {8n + 3 : n E N), and suppose that we define the addition + and multiplication . operations on this set in the usual way. Which of the following properties does this number system satisfy for all a, b, c E S? O (P1) a + (b + c) = (a + b) + c (associative law of addition) (P2) a + 0 = a = 0 + a (existence of additive identity) O (P3) a + (-a) = 0 = (-a) + a (existence of additive inverses) O (P4) a + b = b + a (commutative law of addition) O (P5) a . (b . c) = (a . b) . c (associative law of multiplication) O (P6) a . 1 = a = 1 . a (existence of multiplicative identity) O (P7) a . al = 1 = al . a (existence of multiplicative inverses) O (P8) a . b = b . a (commutative law for multiplication) O (p9) a . (b + c) = a . b + a . c (distributive law) O (P10) either a a (trichotomy law) O (P11) a + b E S (closure under addition) O (P12) a . b E S (closure under multiplication)