## Mathematical Systems Exercise

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?

 • 1 2 3 1 1 2 3 2 2 0 2 3 3 2 1

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?

 • G N U G N U G N U G N U G N U

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

 + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 addition modulo 4
 + 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 multiplication modulo 4

a. Use the tables to calculate (modulo 4):

(i) 2 + 1
(ii) 3 × 2

b. 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.

a. Which is the identity element?

b. What is the inverse of J?

c. Find x so that P * x = J

 * Q Z J P Q J Q P Z Z Q Z J P J P J Z Q P Z P Q J multiplication modulo 4

18. If a * b means √(a2 + b2)

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 8

a. 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 is g.

Find :
(i) e * f
(ii) h * g