|
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
