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