Modulo arithmetic For two integers ‘a’ and ‘b’ and a positive integer ‘n’, let + represent the operation a + b = (a+b)mod(n), that is, the result of a + b is the remainder of the usual sum a + b after dividing by n (using a natural representation). Similarly, let * represent the operation a * b = (a*b)mod(n), that is, the result of a * b is the remainder of the product a*b after dividing by n. For example using the set S = {2, 5, 8} and n = 9, 5 + 8 = 4, since 4 is the remainder after dividing 13 (5 + 8) by 9; and 2 * 8 = 7, since 7 is the remainder after dividing 16 (2*8) by 9. a. For each set S and number n specified below, create two “operation” tables (these are called Cayley tables), one showing the results for the operation +, and one showing the results of the operation * for each pair of elements from S (see the example tables in the notes). i. S1 = {0, 1} and n = 2 ii. S2 = {1, 2} and n = 3 iii. S3 = {0, 2, 4, 6} and n = 8 iv. S4 = {1, 3, 5, 7} and n = 8 v. S5 = {1, 2, 3, 4} and n = 5 vi. S6 = {1, 2, 4, 5, 7, 8} and n = 9
Modulo arithmetic
For two integers ‘a’ and ‘b’ and a positive integer ‘n’, let + represent the
operation a + b = (a+b)mod(n), that is, the result of a + b is the remainder of
the usual sum a + b after dividing by n (using a natural representation).
Similarly, let * represent the operation a * b = (a*b)mod(n), that is, the
result of a * b is the remainder of the product a*b after dividing by n. For
example using the set S = {2, 5, 8} and n = 9, 5 + 8 = 4, since 4 is the
remainder after dividing 13 (5 + 8) by 9; and 2 * 8 = 7, since 7 is the
remainder after dividing 16 (2*8) by 9.
a. For each set S and number n specified below, create two “operation”
tables (these are called Cayley tables), one showing the results for the
operation +, and one showing the results of the operation * for each pair
of elements from S (see the example tables in the notes).
i. S1 = {0, 1} and n = 2
ii. S2 = {1, 2} and n = 3
iii. S3 = {0, 2, 4, 6} and n = 8
iv. S4 = {1, 3, 5, 7} and n = 8
v. S5 = {1, 2, 3, 4} and n = 5
vi. S6 = {1, 2, 4, 5, 7, 8} and n = 9
a. Here are the Cayley tables for each set S and number n:
i. S1 = {0, 1} and n = 2
+ | 0 1
--|----
0 | 0 1
1 | 1 0
* | 0 1
--|----
0 | 0 0
1 | 0 1
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b. Identify the systems from task (2a) that demonstrate closure, i.e., the
result of the operation always produces an element in the set.
c. Identify the systems from task (2a) that have an identity element.
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