The problem defines a binary operation $*$ on the set $S = \{0, 2, 4, 6, 8, 10\}$ as $x * y = x + y - xy$. We need to compute $0 * 2$, $4 * 6$, $0 * 8$, and $2 * 8$. Also, we need to determine if the set $S$ is closed under the operation $*$.

1. Problem Description

The problem defines a binary operation $*$ on the set $S = \{0, 2, 4, 6, 8, 10\}$ as $x * y = x + y - xy$. We need to compute $0 * 2$, $4 * 6$, $0 * 8$, and $2 * 8$. Also, we need to determine if the set $S$ is closed under the operation $*$.

2. Solution Steps

(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 check if $x * y$ is always in $S$ when $x$ and $y$ are in $S$. From the calculations above, we have:

$0 * 2 = 2$, which is in $S$.

$4 * 6 = -14$, which is not in $S$.

$0 * 8 = 8$, which is in $S$.

$2 * 8 = -6$, which is not in $S$.

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

3. Final Answer

(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 $*$.