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 h is g.

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