In set theory and algebra, the statement "Let $R$ be an equivalence relation" means that $R$ is a relation that satisfies the following three properties on a set $A$:

1. Reflexivity: For every $a \in A$, $aRa$ (i.e., $a$ is related to itself).
2. Symmetry: If $aRb$, then $bRa$ (i.e., if $a$ is related to $b$, then $b$ is related to $a$).
3. Transitivity: If $aRb$ and $bRc$, then $aRc$ (i.e., if $a$ is related to $b$ and $b$ is related to $c$, then $a$ is related to $c$).

Now, regarding the expression $a \in [a]$:

• Equivalence Class: Given an equivalence relation $R$, the equivalence class of an element $a \in A$, denoted by $[a]$, is the set of all elements in $A$ that are related to $a$ under the relation $R$. Formally, $[a] = \{ x \in A \mid xRa \}$ In other words, $[a]$ consists of all elements that are equivalent to $a$ under the relation $R$.

• Interpretation of $a \in [a]$: The statement $a \in [a]$ simply means that an element $a$ is always a member of its own equivalence class. This follows directly from the reflexivity property of equivalence relations, which ensures that $aRa$, meaning $a$ is related to itself.

Thus, any element is always in its own equivalence class, so $a \in [a]$ is universally true for any element $a$ under the equivalence relation $R$.

Would you like more details or further clarifications?

Here are 5 related questions to explore:

1. How do we construct equivalence classes for a specific equivalence relation?
2. What are some examples of equivalence relations in real-life scenarios?
3. How does the concept of partitions of a set relate to equivalence relations?
4. How can equivalence classes be used in modular arithmetic?
5. What is the significance of equivalence relations in group theory or topology?

Tip: When working with equivalence relations, remember that the set of all equivalence classes forms a partition of the underlying set. This means that every element of the set belongs to exactly one equivalence class.