A relation R is defined on the set A = \1, 2, 3, 4, 5\ by a R b a + b = 2n, n N. List all the equivalence classes of A under R.

The condition $a+b = 2n, n \in \mathbb{N}$ means that $a+b$ must be an even positive integer. Since $a, b \in \{1, 2, 3, 4, 5\}$, the smallest sum is $1+1=2$ and the largest is $5+5=10$. All these sums are positive.
Therefore, the condition simplifies to $a+b$ is an even number. This implies that $a$ and $b$ must have the same parity (both even or both odd).

• Reflexive: For any $a \in A$, $a+a = 2a$. Since $a$ is an integer, $2a$ is always an even number. Thus, $a R a$ for all $a \in A$.
• Symmetric: If $a R b$, then $a+b$ is an even number. Since addition is commutative, $b+a$ is also an even number. Thus, $b R a$.
• Transitive: If $a R b$ and $b R c$, then $a+b$ is even and $b+c$ is even. If $a+b$ is even, $a$ and $b$ have the same parity. If $b+c$ is even, $b$ and $c$ have the same parity. Therefore, $a$, $b$, and $c$ must all have the same parity. If $a$ and $c$ have the same parity, then $a+c$ is an even number. Thus, $a R c$. Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.

An equivalence class $[x]$ consists of all elements $y \in A$ such that $y R x$. This means $y$ and $x$ must have the same parity.

• For $x=1$ (an odd number): $[1] = \{y \in A \mid y \text{ is odd}\} = \{1, 3, 5\}$.

• For $x=2$ (an even number): $[2] = \{y \in A \mid y \text{ is even}\} = \{2, 4\}$.

• For $x=3$ (an odd number): $[3] = \{y \in A \mid y \text{ is odd}\} = \{1, 3, 5\}$. (Same as $[1]$)

• For $x=4$ (an even number): $[4] = \{y \in A \mid y \text{ is even}\} = \{2, 4\}$. (Same as $[2]$)

• For $x=5$ (an odd number): $[5] = \{y \in A \mid y \text{ is odd}\} = \{1, 3, 5\}$. (Same as $[1]$)

The distinct equivalence classes are the sets of odd numbers and even numbers within $A$.

The equivalence classes of $A$ under $R$ are $\boxed{\{1, 3, 5\} \text{ and } \{2, 4\}}$.