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Class 12 RD Sharma Solutions – Chapter 3 Binary Operations – Exercise 3.5

  • Last Updated : 28 Dec, 2020

Question 1. Construct the composition table for ×4 on set S = {0,1,2,3}.

Solution:

Given that × on set S = {0, 1, 2, 3}

We have to find the composition table for ×4 by using multiplication and modulo operations.

As 0 x4 0 = 0 (0 × 0 = 0 and 0 modulo 4 = 0 % 4 = 0)

As 1 x4 1 = 1 (1 × 1 = 1 and 1 modulo 4 = 1 % 4 =1)

As 1 x4 2 = 2 (1 × 2 = 2 and 2 modulo 4 = 2 % 4 = 2)

As 3 x4 1 = 3 (3 × 1 = 3 and 3 modulo 4 = 3 % 4 = 3)

As 4 x4 4 = 0 (2 × 2 = 4 and 4 modulo 4 = 4 % 4 = 0)

.

So on. . .

Here % – (modulo operator) means it gives the remainder of the number when divided by the other number (i.e., 4)

Therefore, the composition table is:

x40123
00000
10123
20202
30321

Question 2. Construct the composition table for +5 on set S = {0, 1, 2, 3, 4}.

Solution:

Given that +5  on set S = {0,1,2,3,4}

We have to find the composition table for +5 by using addition and modulo operations.

As 0 +5 0 = 0 (0 + 0 = 0 and 0 modulo 5 = 0 % 5 = 0)

As 1 +5 0 = 1 (1 + 0 = 1 and 1 modulo 5 = 1 % 5 = 1)

As 1 +5 1 = 2 (1 + 1 = 2 and 2 modulo 5 = 2 % 5 =2)

As 1 +5 2 = 3 (1 + 2 = 3 and 3 modulo 5 = 3 % 5 = 3)

As 3 +5 1 = 4 (3 + 1 = 4 and 4 modulo 5 = 4 % 5 = 4)

As 2 +5 2 = 4 (2 + 2 = 4 and 4 modulo 5 = 4 % 5 = 4)

.

So on . . .

Here % – (modulo operator) means it gives the remainder of the number when divided by the other number (i,e., 5)

Therefore, the composition table is:

+501234
001234
112340
223401
334012
440123

Question 3. Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.

Solution:

Given that ×6  on set S = {0, 1, 2, 3, 4, 5}

We have to find the composition table for ×6 by using multiplication and modulo operations.

As 0 x6 0 = 0 (0 × 0 = 0 and 0 modulo 6 = 0 % 6 = 0)

As 1 x6 1 = 1 (1 × 1 = 1 and 1 modulo 6 = 1 % 6 =1)

As 1 x6 2 = 2 (1 × 2 = 2 and 2 modulo 6 = 2 % 6 = 2)

As 3 x6 1 = 3 (3 × 1 = 3 and 3 modulo 6 = 3 % 6 = 3)

As 2 x6 2 = 4 (2 × 2 = 4 and 4 modulo 6 = 4 % 6 = 4)

As 5 x6 1 = 5 (5 × 1 = 5 and 5 modulo 6 = 5 % 6 = 5)

.

so on . . .

Here % – (modulo operator) means it gives the remainder of the number when divided by the other number (i.e., 6)

Therefore, the composition table is :

x6012345
0000000
1012345
2024024
3030303
4042042
5054321

Question 4. Construct the composition table for ×5 on set S = {0, 1, 2, 3, 4}.

Solution:

Given that ×5  on set S = {0,1,2,3,4}

We have to find the composition table for ×5 by using multiplication and modulo operations.

As 0 x5 0 = 0 (0 × 0 = 0 and 0 modulo 5 = 0 % 5 = 0)

As 1 x5 1 = 1 (1 × 1 = 1 and 1 modulo 5 = 1 % 5 =1)

As 1 x5 2 = 2 (1 × 2 = 2 and 2 modulo 5 = 2 % 5 = 2)

As 2 x5 2 = 4 (2 × 2 = 4 and 4 modulo 5 = 4 % 5 = 4)

As 3 x5 3 = 2 (3 × 4 = 12 and 12 modulo 5 = 12 % 5 = 2)

.

So on . . .

Here % – (modulo operator) means it gives the remainder of the number when divided by the other number (i.e., 5)

Therefore, the composition table is:

x501234
000000
101234
202413
303142
404321

Question 5. Construct the composition table for ×10 on set S = {1, 3, 7, 9}.

Solution:

Given that ×10  on set S = {1,3,7,9}

We have to find the composition table for ×10 by using multiplication and modulo operations.

As 1 x10 1 = 1 (1 × 1 = 1 and 1 modulo 10 = 1 % 10 = 1)

As 1 x10 7 = 7 (1 × 7 = 7 and 7 modulo 10 = 2 % 10 = 7)

As 3 x10 1 = 3 (3 × 1 = 3 and 3 modulo 10 = 3 % 10 = 3)

As 9 x10 7 = 3 (9 × 7 = 63 and 63 modulo 10 = 63 % 10 = 3)

.

So on . . .

Here % – (modulo operator) means it gives the remainder of the number when divided by the other number (i.e., 10)

Therefore, the composition table is:

x101379
11379
33917
77193
99731

From the composition table, we can observe that elements 1 multiples (i.e., 1st row) are same as top most row.

Therefore, 1 ∈ S is the identity element for x10.

We have to find the inverse of 3:

As 3 x10 7 = 1 (3 × 7 = 21 and 21 modulo 10 = 1) in the composition table.

So the inverse of 3 is 7

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