The relation $R = \{(x, y): x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:

Step 1: Check for Reflexivity

A relation $R$ on a set $A$ is reflexive if $(x, x) \in R$ for every $x \in A$. In this case, we need to check if $x + x$ is always even for any integer $x$. The sum $x + x$ simplifies to $2x$.

Step 2: Evaluate Reflexivity

Since $2x$ is a multiple of 2, it is always an even number for any integer $x$. Therefore, $(x, x) \in R$ for all $x \in \mathbb{Z}$, which means the relation $R$ is reflexive.

Step 3: Check for Symmetry

A relation $R$ is symmetric if whenever $(x, y) \in R$, then $(y, x) \in R$. This means if $x + y$ is even, we need to check if $y + x$ is also even. Since addition of integers is commutative, $x + y = y + x$.

Step 4: Evaluate Symmetry

Since $x + y = y + x$, if $x + y$ is even, then $y + x$ must also be even. Thus, $(y, x) \in R$ whenever $(x, y) \in R$, which means the relation $R$ is symmetric.

Step 5: Check for Transitivity

A relation $R$ is transitive if whenever $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$. This means if $x + y$ is even and $y + z$ is even, we need to check if $x + z$ is also even.

Step 6: Evaluate Transitivity

If $x + y$ is even, then $x$ and $y$ must have the same parity (both even or both odd). Similarly, if $y + z$ is even, then $y$ and $z$ must have the same parity. If $x$ and $y$ have the same parity, and $y$ and $z$ have the same parity, it implies that $x$ and $z$ must also have the same parity. When $x$ and $z$ have the same parity, their sum $x + z$ will always be even. For example, if $x$ is odd and $y$ is odd, $x+y$ is even. If $y$ is odd and $z$ is odd, $y+z$ is even. Then $x$ is odd and $z$ is odd, so $x+z$ is even. Thus, the relation $R$ is transitive.

Step 7: Conclusion

Since the relation $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.

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