To prove that $a \in [a]$ for an equivalence relation $R$ on a set $A$, let's break it down step by step.

Definitions and Notations:

1. Let $R$ be an equivalence relation on a set $A$. An equivalence relation is a binary relation that is:

   • Reflexive: $aRa$ for all $a \in A$
   • Symmetric: If $aRb$, then $bRa$ for all $a, b \in A$
   • Transitive: If $aRb$ and $bRc$, then $aRc$ for all $a, b, c \in A$

2. The equivalence class of an element $a \in A$ under the relation $R$ is denoted by $[a]$, and it is defined as: $[a] = \{ x \in A \mid aRx \}$ This is the set of all elements of $A$ that are related to $a$ under the relation $R$.

Proof:

We need to prove that: $a \in [a]$ This means we must show that $a$ is related to itself, i.e., $aRa$.

By the definition of an equivalence relation, one of the key properties of $R$ is reflexivity. Reflexivity states that for all elements $a \in A$, the relation $aRa$ holds. In other words, each element is related to itself.

Since $R$ is reflexive, we have $aRa$. This implies that: $a \in \{ x \in A \mid aRx \} = [a]$

Thus, $a \in [a]$.

Conclusion:

By the reflexive property of the equivalence relation $R$, $a \in [a]$ for any $a \in A$.

Let me know if you'd like more details or have further questions.

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Here are 5 related questions for further exploration:

1. What is an example of an equivalence relation on the set of integers?
2. How do symmetric and transitive properties contribute to an equivalence relation?
3. Can an equivalence class contain more than one element?
4. How do equivalence classes partition a set?
5. What is the difference between an equivalence relation and a partial order?

Tip: Always check for the reflexive, symmetric, and transitive properties to verify whether a given relation is an equivalence relation.