To analyze whether the relation $R$ defined by $aRb$ if and only if $2a + 5b$ is divisible by 7 is an equivalence relation on $\mathbb{Z}$ (the set of integers), we need to check if it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

1. Reflexivity

A relation $R$ on a set $S$ is reflexive if every element is related to itself. That is, $aRa$ for all $a \in \mathbb{Z}$.

For $aRa$, we need $2a + 5a$ to be divisible by 7. Simplifying:

$2a + 5a = 7a$

Since $7a$ is clearly divisible by 7 for any integer $a$, the relation $R$ is reflexive.

2. Symmetry

A relation $R$ on a set $S$ is symmetric if whenever $aRb$, then $bRa$ must also hold for all $a, b \in \mathbb{Z}$.

Suppose $aRb$ holds. Then, $2a + 5b$ is divisible by 7. We need to check if $bRa$ also holds, which means $2b + 5a$ should be divisible by 7.

Let's consider $2a + 5b \equiv 0 \mod 7$.

This implies: $2a + 5b \equiv 0 \mod 7$

Now, consider $2b + 5a$: $2b + 5a = 5a + 2b$

Given $2a + 5b \equiv 0 \mod 7$, we want to check if $5a + 2b$ is also congruent to 0 modulo 7. Notice that: $2(5a + 2b) \equiv 5(2a + 5b) \equiv 0 \mod 7$

This simplification confirms that $2b + 5a$ is indeed divisible by 7, so $bRa$ holds whenever $aRb$. Hence, the relation $R$ is symmetric.

3. Transitivity

A relation $R$ on a set $S$ is transitive if whenever $aRb$ and $bRc$, then $aRc$ must also hold for all $a, b, c \in \mathbb{Z}$.

Assume $aRb$ and $bRc$. Then:

$2a + 5b \equiv 0 \mod 7 \quad \text{and} \quad 2b + 5c \equiv 0 \mod 7$

Adding these, we get: $(2a + 5b) + (2b + 5c) \equiv 0 + 0 \equiv 0 \mod 7$

Simplifying: $2a + 7b + 5c \equiv 2a + 5c \equiv 0 \mod 7$

Thus, $2a + 5c$ is divisible by 7, so $aRc$ holds. Therefore, the relation $R$ is transitive.

Conclusion: Equivalence Relation

Since $R$ is reflexive, symmetric, and transitive, $R$ is an equivalence relation on $\mathbb{Z}$.

Anti-Symmetry

A relation $R$ on a set $S$ is anti-symmetric if $aRb$ and $bRa$ imply $a = b$.

Since $R$ is symmetric, this implies that if $aRb$, then $bRa$ also holds, regardless of whether $a = b$. For example, $a = 0$ and $b = 0$ clearly satisfy $aRb$, but so does any pair $(a, b)$ where $2a + 5b \equiv 0 \mod 7$.

This means $R$ is not anti-symmetric because symmetry and anti-symmetry are mutually exclusive unless $a = b$.

Summary

• The relation $R$ is an equivalence relation on $\mathbb{Z}$.
• The relation $R$ is not anti-symmetric.

Would you like more details or have any other questions?

Here are 8 related questions:

1. Can you provide an example where a relation is symmetric but not anti-symmetric?
2. What are the implications of a relation being reflexive, symmetric, and transitive?
3. How would the relation change if the coefficients in the definition of $R$ were different?
4. What are some other examples of equivalence relations in mathematics?
5. Can a relation be anti-symmetric and symmetric at the same time? If so, under what conditions?
6. How does the concept of modular arithmetic help in proving symmetry and transitivity?
7. Is the set of equivalence classes under this relation finite or infinite?
8. How would you describe the equivalence classes for this relation?

Tip: When analyzing a relation, it's useful to test it against simple cases (like 0 or 1) to gain intuition before generalizing.