The given question defines a relation $R$ from set $A$ to set $B$, where the sets are:

• $Y = \{ 0, 1, 2 \}$
• $Z = \{ 0, 1 \}$

For any ordered pair $(x, y)$ where $x \in Y$ and $y \in Z$, the relation $(x, y) \in R$ is defined as: $\frac{x + y}{2} \text{ is an integer}.$

Let me break down the questions one by one:

a. State explicitly which ordered pairs are in $Y \times Z$ and which are in $R$.

First, let's identify all ordered pairs in $Y \times Z$ (the Cartesian product): $Y \times Z = \{(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)\}$

Now, apply the condition $\frac{x + y}{2}$ to see which pairs belong to $R$:

1. For $(0, 0)$: $\frac{0 + 0}{2} = 0$, which is an integer, so $(0, 0) \in R$.
2. For $(0, 1)$: $\frac{0 + 1}{2} = 0.5$, which is not an integer, so $(0, 1) \notin R$.
3. For $(1, 0)$: $\frac{1 + 0}{2} = 0.5$, which is not an integer, so $(1, 0) \notin R$.
4. For $(1, 1)$: $\frac{1 + 1}{2} = 1$, which is an integer, so $(1, 1) \in R$.
5. For $(2, 0)$: $\frac{2 + 0}{2} = 1$, which is an integer, so $(2, 0) \in R$.
6. For $(2, 1)$: $\frac{2 + 1}{2} = 1.5$, which is not an integer, so $(2, 1) \notin R$.

Thus, the pairs that are in $R$ are: $R = \{ (0, 0), (1, 1), (2, 0) \}$

b. Is $1 \, R \, 0$? Is $2 \, R \, 0$? Is $2 \, R \, 1$?

To determine this, we check if the pairs $(1, 0)$, $(2, 0)$, and $(2, 1)$ belong to $R$ based on the condition $\frac{x + y}{2}$:

• $1 \, R \, 0$? For $(1, 0)$: $\frac{1 + 0}{2} = 0.5$, so No.
• $2 \, R \, 0$? For $(2, 0)$: $\frac{2 + 0}{2} = 1$, so Yes.
• $2 \, R \, 1$? For $(2, 1)$: $\frac{2 + 1}{2} = 1.5$, so No.

c. What are the domain and co-domain of $R$?

• Domain: The domain of $R$ is the set of $x$ values from the ordered pairs in $R$. The pairs in $R$ are $(0, 0)$, $(1, 1)$, and $(2, 0)$. So, the domain is: $\text{Domain of } R = \{ 0, 1, 2 \}$

• Co-domain: The co-domain of $R$ is the set of $y$ values from the ordered pairs in $R$. The pairs in $R$ are $(0, 0)$, $(1, 1)$, and $(2, 0)$. So, the co-domain is: $\text{Co-domain of } R = \{ 0, 1 \}$

Would you like further details or have any questions? Here are 5 related questions for further practice:

1. Can a pair like $(3, 1)$ be considered for $R$?
2. What is the range of the relation $R$?
3. How would the relation change if $Z = \{0, 1, 2\}$?
4. What is the inverse of the relation $R$?
5. What happens to the relation if we change the condition to $\frac{x - y}{2}$?

Tip: When dealing with relations, always verify the condition of the relation for each pair systematically to ensure accuracy.