Let's go through each part of this problem, step-by-step.

Given Information:

• Set $A = \{1, 2, 3, 4, 5, 6\}$.
• Relation $R$ on $A$ is defined such that $mRn$ if and only if $2 \mid (m + n)$, which means that $m + n$ is even.

Solution:

1. (a) Directed Graph of $R$: To construct the directed graph, we need to determine which pairs $(m, n)$ satisfy the relation $R$.

   • For two elements $m, n \in A$, the sum $m + n$ is even if both $m$ and $n$ are either both even or both odd.
   • Even numbers in $A$: $2, 4, 6$
   • Odd numbers in $A$: $1, 3, 5$

   Therefore:

   • $mRn$ holds for pairs $(m, n)$ where both $m$ and $n$ are either even or both are odd.

   Pairs that satisfy $R$:

   • For odd elements $\{1, 3, 5\}$: pairs are $(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5)$.
   • For even elements $\{2, 4, 6\}$: pairs are $(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)$.

   Using these pairs, you can create a directed graph with nodes labeled $1$ to $6$ and arrows connecting nodes according to the pairs above.

2. (b) Equivalence Class $[5]_R$: The equivalence class $[5]_R$ consists of all elements $n \in A$ such that $5Rn$.

   Since $5$ is odd, we need to find all odd elements in $A$. These are $1, 3,$ and $5$.

   Thus, $[5]_R = \{1, 3, 5\}$.

3. (c) Number of Distinct Equivalence Classes: We observe that $R$ groups elements into two classes based on parity:

   • Odd elements: $\{1, 3, 5\}$
   • Even elements: $\{2, 4, 6\}$

   These two groups represent two distinct equivalence classes under $R$.

   Therefore, the relation $R$ has 2 distinct equivalence classes.

Would you like further details on any part of the solution?

Related Questions

1. What makes a relation an equivalence relation, and how can you check it?
2. How do you interpret equivalence classes in terms of partitions of a set?
3. How would the solution change if $A$ had a different number of elements?
4. Can you find equivalence classes in non-numeric sets, such as sets of words or other objects?
5. How would you generalize this relation for any finite set $A$ with an arbitrary rule for even sums?

Tip

To verify that a relation is an equivalence relation, always check that it is reflexive, symmetric, and transitive. These properties are essential for defining equivalence classes.