1 3 + (5 + 2) = (3 + 5) + 2 associative 2 5 + 4 = 4 + 5 commutative 3 3 × 1/3 = 1 inverse 4 4 + 0 = 4 identity 5 3(x + 2) = 3x + 6 distributative 6 3 × 5 = 5 × 3 commutative 7 4 × 1 = 4 identity 8 = distributative 9 4 + -4 = 0 inverse 10 (2 × 5) × 4 = 2 × (5 × 4) associative

11. a. Yes, the system is closed.
b.The identity element is 1.
c. The inverse of 1 is 1.
d. No.
e. 1 × (-1 × 1) = (1 × -1) × 1 is an example of the associative property.

12. a. Yes, the set closed under +?
b. The identity element is E?
c. The inverse of E under addition is E?
d. Yes
e. Associative.

13. a.The identity element is 1.
b.Because 2 • 2 = 0 and 0 is not part of the system.
c. 2 does not have an inverse.
d. No, because every element does not have an inverse.

14. a. The identity element is U.
b. The inverse of G is N.
c. G * N = U.

15. a. (i) 2 + 1 = 3
(ii) 3 × 2 = 2

b. (i) 3 × (2 + 1) = 1
(ii) (3 × 2) + (3 × 1) = 1
This shows the distributive property.
c. The multiplication table does not show a group as 0 does not have an inverse.

16. a. a * i = a.
b. The inverse of a is b.
c. a * b = i.
d. The inverse of c is c.

17. a. The identity element is Z.
b. The inverse of J is J
c. x = P

18. a. 3 * 4 = 5
b. 8 * 6 = 10

19.

 x 2 4 6 8 2 4 8 2 6 4 8 6 4 2 6 2 4 6 8 8 6 2 8 4

a. The set is closed.
b. The system is commutative.
c. The identity element is 6
d. The inverse of 6 is 6.
e. x = 8

20. (i) e * f = f
(ii) h * g = e