To prove that $R$ is an equivalence relation on the set of integers, we need to verify that $R$ satisfies the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

Given relation $R$ is defined such that $(a, b) R (c, d)$ if and only if $3a + 4b = 3c + 4d$.

1. Reflexivity: We need to show that $(a, b) R (a, b)$ for any integer $a$ and $b$.

Consider $(a, b)$. We need to check if $(a, b) R (a, b)$. According to the definition, this requires: $3a + 4b = 3a + 4b$

This equation is clearly true.

Thus, $(a, b) R (a, b)$ and $R$ is reflexive.

2. Symmetry: We need to show that if $(a, b) R (c, d)$, then $(c, d) R (a, b)$.

Assume $(a, b) R (c, d)$. By the definition of $R$, this means: $3a + 4b = 3c + 4d$

To show symmetry, we need to check if: $3c + 4d = 3a + 4b$

This is obviously true because equality is symmetric.

Thus, $(c, d) R (a, b)$ and $R$ is symmetric.

3. Transitivity: We need to show that if $(a, b) R (c, d)$ and $(c, d) R (e, f)$, then $(a, b) R (e, f)$.

Assume $(a, b) R (c, d)$ and $(c, d) R (e, f)$. By the definition of $R$, we have: $3a + 4b = 3c + 4d$ $3c + 4d = 3e + 4f$

To show transitivity, we need to check if: $3a + 4b = 3e + 4f$

Since $3a + 4b = 3c + 4d$ and $3c + 4d = 3e + 4f$, we can deduce that: $3a + 4b = 3e + 4f$

Thus, $(a, b) R (e, f)$ and $R$ is transitive.

Since $R$ is reflexive, symmetric, and transitive, $R$ is an equivalence relation on the set of integers.

Would you like more details on any of these steps or have any questions?

Here are 8 related questions to expand on this topic:

1. Can you provide an example of equivalence classes formed by this relation?
2. How does the structure of equivalence classes help in simplifying complex problems?
3. How can equivalence relations be visualized using graphs or diagrams?
4. What are some applications of equivalence relations in abstract algebra?
5. How do equivalence relations relate to partitions of a set?
6. Can you show how this equivalence relation could be used to group integers in a practical problem?
7. How do equivalence relations differ from congruence relations in modular arithmetic?
8. What are the implications of equivalence relations in defining quotient sets?

Tip: When proving properties of relations, carefully examine each property (reflexivity, symmetry, transitivity) and use the given definitions directly to show that they hold. This systematic approach ensures thorough verification.