Let's address each part of the problem systematically.

Part (a): Which of the following are in the relation?

To determine if a pair $(m, n)$ is in the relation $R$, we check if $m - n$ is odd. If $m - n$ is odd, then $(m, n) \in R$.

1. $(8, 2)$: $8 - 2 = 6 \quad (\text{even, not odd})$ So, $(8, 2) \notin R$.

2. $(1, 4)$: $1 - 4 = -3 \quad (\text{odd})$ So, $(1, 4) \in R$.

3. $(5, -3)$: $5 - (-3) = 5 + 3 = 8 \quad (\text{even, not odd})$ So, $(5, -3) \notin R$.

4. $(3, 2)$: $3 - 2 = 1 \quad (\text{odd})$ So, $(3, 2) \in R$.

Conclusion for Part (a): The pairs in the relation $R$ are: $(1, 4) \quad \text{and} \quad (3, 2)$

Part (b): List five integers that are related by $R$ to $5$.

To find integers $n$ such that $(5, n) \in R$, we need $5 - n$ to be odd. For this to happen, $n$ must be even (since $5$ is odd, subtracting an even integer from it will yield an odd result).

Some examples of even integers related to $5$ are: $n = 0, 2, 4, 6, 8$

Therefore, five integers that are related to $5$ are: $0, 2, 4, 6, 8$

Part (c): Prove that if $n$ is any even integer, then $(n, 3) \in R$.

To prove this, assume $n$ is even. By definition, an even integer can be written as $n = 2k$ for some integer $k$.

Now, consider $n - 3$: $n - 3 = 2k - 3$ Since $2k$ is even and $3$ is odd, the difference $2k - 3$ is odd (the difference between an even and an odd number is odd).

Therefore, $n - 3$ is odd, which means $(n, 3) \in R$ for any even $n$.

Part (d): Is $R$ an equivalence relation? If so, how many equivalence classes does it have?

To determine if $R$ is an equivalence relation on $\mathbb{Z}$, it must satisfy three properties: reflexivity, symmetry, and transitivity.

1. Reflexivity: $R$ is not reflexive because $(m, m) \in R$ would require $m - m = 0$, which is not odd. Therefore, $R$ does not satisfy reflexivity.

2. Symmetry: $R$ is symmetric because if $(m, n) \in R$, then $m - n$ is odd, which implies that $n - m$ (the opposite sign) is also odd. Thus, if $(m, n) \in R$, then $(n, m) \in R$, so $R$ is symmetric.

3. Transitivity: $R$ is not transitive. For example, consider $(1, 4) \in R$ and $(4, 3) \in R$ (since $1 - 4 = -3$ and $4 - 3 = 1$, both of which are odd). However, $(1, 3) \notin R$ since $1 - 3 = -2$ (an even number).

Since $R$ fails to be reflexive and transitive, $R$ is not an equivalence relation.

Summary of Answers

1. Part (a): The pairs $(1, 4)$ and $(3, 2)$ are in the relation.
2. Part (b): Five integers related to $5$ are $0, 2, 4, 6,$ and $8$.
3. Part (c): If $n$ is even, then $(n, 3) \in R$.
4. Part (d): $R$ is not an equivalence relation.

Do you want further explanation on any part, or have additional questions?

Here are 5 related questions you might find helpful:

1. How do we formally define an equivalence relation and its properties?
2. What examples of relations can be equivalence relations in the set of integers?
3. How can we determine the number of equivalence classes if a relation is an equivalence relation?
4. What are some methods to check if a relation is reflexive, symmetric, or transitive?
5. Can we visualize relations on integers using a graph or matrix approach?

Tip: When testing whether a relation is transitive, consider finding three elements that satisfy $(a, b) \in R$ and $(b, c) \in R$ but not $(a, c)$.