Page 40 - 11-Math-2 Sets Functions and Groups

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of 5 we insert 1( = 5#4 ) in the table and in place of 1 + 3 = 4 , we insert 0(= 4#4).

Example 8: Give the table for multiplication of elemnts of the set of residue classes modulo

4.

Solution: Clearly {0,1,2,3} is the set of residues that we have to - 0 1 23

consider. We multiply pairs of elements as in ordinary 0 0 0 00

multiplcation except that when the product equals or exceeds 4, 1 0 1 2 3
we divide it out by 4 and insert the remainder only in the table.
2 0 2 02
Thus 3%2=6 but in place of 6 we insert 2 (= 6#4 )in the table and 3 0 3 2 1
in place of 2%2=4, we insert 0(= 4#4).

Example 9: Give the table for multiplication of elements of the set of residue classes modulo
8.

Solution: Table is given below:

-0 1 2 3 4 5 6 7

00 0 0 0 0 0 0 0

10 1 2 3 4 5 6 7
20 2 4 6 0 2 4 6
30 3 6 1 4 7 2 5
40 4 0 4 0 4 0 4
50 5 2 7 4 1 6 3
60 6 4 2 0 6 4 2
70 7 6 5 4 3 2 1

Note: For performing multiplication of residue classes 0 is generally omitted.

2.12.1 Properties of Binary Operations

Let S be a non-empty set and %.... a binary operation on it. Then %.... may possess one or

more of the following properties: -

i) Commutativity: %.... is said to be commutative if

a %.... b = b %.... a " a,bd S.

version: 1.1

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