State which properties of mathematical systems, chosen from this list, are shown by the following equations.

commutative
 associative
 distributive
 identitiy element
 inverse
1.

3 + (5 + 2)

=

(3 + 5) + 2

2.

5 + 4

=

4 + 5

3.

3 × 1/3

=

1

4.

4 + 0

=

4

5.

3(x + 2)

=

3x + 6

6.

3 × 5

=

5 × 3

7.

4 × 1

=

4

8.

=


9.

4 + 4

=

0

10.

(2 × 5) × 4

=

2 × (5 × 4)

11. Consider the system {1, 0, 1} under multiplication.
a. Is the system closed under multiplication?
b. What is the identity element?
c. What is the inverse of 1?
d. Does 0 have an inverse?
e. Give one example to show that the associative law is true for the system.
12. Consider the system ({E, O}, +) where E is any even number and O is any odd number.
a. Is the set closed under +?
b. What is the identity element?
c. What is the inverse of E under addition?
d. Does E + (O + E) = (E + O) + E?
e. What property does this show?
13. The table below is for multiplication modulo 4 of the whole numbers {1, 2, 3}
a. Name the identity element. b. Explain why {1, 2, 3} under the operation above is not closed. c. Which elements of {1, 2, 3} do not have inverses? d. Is the system ({1, 2, 3},•) a group? 

14. {G, N, U} with the operation * forms a group as defined by the table:
a. Name the identity element. b. Give the inverse of G. c. What is the value of G * N?


15. Here are the tables for addition and multiplication in "clock arithmetic" modulo 4.


a. Use the tables to calculate (modulo 4):
(i) 2 + 1
(ii) 3 × 2b. Use the tables to calculate (modulo 4):
(i) 3 × (2 + 1)
(ii) (3 × 2) + (3 × 1)
What rule or law do these last two calculations test?c. Which of the tables above does not represent a group. Give a brief reason.
16. The set {a, b, c, i} with the binary operation * is a group. The identity is i.
a. Simplify a * i
The inverse of b is a.
b. What is the inverse of a?
c. Simplify a * b.
d. What is the inverse of c?
17. The table below is the table of a group.


18. If a * b means √(a^{2} + b^{2})
a. Calculate 3 * 4
b. Calculate 8 * 6
19. Complete the table below for the set {2, 4, 6, 8} under multiplication, modulo 10.
x 2 4 6 8 2 4 6 8a. Is the set closed?
b. Is the system commutative?
c. What is the identity element?
d. What is the inverse of 6?
e. Solve the equation 2 × x = 6
20. The set {e, f, g, h} forms a group under an operation *. The identity element is e. The inverse of h is g.
Find :
(i) e * f
(ii) h * g