Name: Properties of Binary Operations
Uploaded: 2019-07-19
Duration: 12 min 41 s
Description: In this video, we will learn about binary operations. We will also understand properties followed by binary operations with the help of suitable examples.

Question

Form the cayley table for the binary operation $*$ defined on the set $S = \{0, 1, 2, 3, 4, 5\}$ by the rule that $a * b = (a + b) (m o d 6)$ and use the table to show that $*$ is commutative. Determine the identity element and $2^{- 1}$ with respect to $*$ $.$

Accepted Answer

Given $S = {0, 1, 2, 3, 4, 5}$ and $a * b = (a + b) (m o d 6)$ $\forall a, b \in S$ .

For $2 * 3 = 5 (m o d 6)$

When we divide $5$ by $6$ , we get the remainder as $5$ , so we have to enter the remainder in the table with respect to the given numbers

$\begin{array}{l} * & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 1 & 2 & 3 & 4 & 5 \\ 1 & 1 & 2 & 3 & 4 & 5 & 0 \\ 2 & 2 & 3 & 4 & 5 & 0 & 1 \\ 3 & 3 & 4 & 5 & 0 & 1 & 2 \\ 4 & 4 & 5 & 0 & 1 & 2 & 3 \\ 5 & 5 & 0 & 1 & 2 & 3 & 4 \end{array}$

Cayley table's values are symmetric along its diagonal axis.

Hence, $*$ is commutative.

For identity: From the table $0 \in S$

$a * 0 = a = 0 * a$ for all $a \in S$

Therefore, $0$ is called identity element

For inverse: $a * a^{- 1} = e = a^{- 1} * a$

$\Rightarrow 2 * 2^{- 1} = 0 = 2^{- 1} * 2$

$\Rightarrow 2 + 2^{- 1} = 0 (m o d 6)$ only when $2^{- 1}$ becomes $4$

Form the cayley table for the binary operation $$ defined on the set $S = \{0, 1, 2, 3, 4, 5\}$ by the rule that $a b = (a + b) (m o d 6)$ and use the table to show that $$ is commutative. Determine the identity element and $2^{- 1}$ with respect to $$ $.$

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Form the cayley table for the binary operation * defined on the set S=0,1,2,3,4,5 by the rule that a*b=a+bmod 6 and use the table to show that * is commutative. Determine the identity element and 2-1 with respect to *.

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Form the cayley table for the binary operation $$ defined on the set $S = \{0, 1, 2, 3, 4, 5\}$ by the rule that $a b = (a + b) (m o d 6)$ and use the table to show that $$ is commutative. Determine the identity element and $2^{- 1}$ with respect to $$ $.$