We are given a set $S = \{0, 2, 4, 6, 8, 10\}$ and an operation $*$ defined on $S$ as $x * y = x + y - xy$. We need to compute the values of $0*2$, $4*6$, $0*8$, and $2*8$. Then we need to determine if the set $S$ is closed under the operation $*$. For the second part, we have a binary operation $\otimes$ defined on the set of real numbers $R - \{0\}$ such that $a \otimes b = 2a + 2b - 5ab$. We need to check if the operation $\otimes$ is commutative.

1. Problem Description

We are given a set $S = \{0, 2, 4, 6, 8, 10\}$ and an operation $*$ defined on $S$ as $x * y = x + y - xy$. We need to compute the values of $0*2$, $4*6$, $0*8$, and $2*8$. Then we need to determine if the set $S$ is closed under the operation $*$.

For the second part, we have a binary operation $\otimes$ defined on the set of real numbers $R - \{0\}$ such that $a \otimes b = 2a + 2b - 5ab$. We need to check if the operation $\otimes$ is commutative.

2. Solution Steps

(1)

(i) $0 * 2 = 0 + 2 - (0)(2) = 2 - 0 = 2$

(ii) $4 * 6 = 4 + 6 - (4)(6) = 10 - 24 = -14$

(iii) $0 * 8 = 0 + 8 - (0)(8) = 8 - 0 = 8$

(iv) $2 * 8 = 2 + 8 - (2)(8) = 10 - 16 = -6$

To determine if the set $S$ is closed under the operation $*$, we need to verify if $x * y \in S$ for all $x, y \in S$.

We have found that $4 * 6 = -14$ and $2 * 8 = -6$. Since $-14$ and $-6$ are not in $S$, the set $S$ is not closed under the operation $*$.

(2)

To check if the operation $\otimes$ is commutative, we need to verify if $a \otimes b = b \otimes a$ for all $a, b \in R - \{0\}$.

$a \otimes b = 2a + 2b - 5ab$

$b \otimes a = 2b + 2a - 5ba$

Since $2a + 2b - 5ab = 2b + 2a - 5ba$, the operation $\otimes$ is commutative.

3. Final Answer

(1)

(i) $0 * 2 = 2$

(ii) $4 * 6 = -14$

(iii) $0 * 8 = 8$

(iv) $2 * 8 = -6$

The set $S$ is not closed under the operation $*$.

(2)

The operation $\otimes$ is commutative.