Let R be the relation on set `A={x:x in Z, 0 le x le 10}` given by `R={(a,b):(a-b) "is divisible by " 4}`. Show that R is an equivalence relation.
Also, write all elements related to 4.

Answer

To show that the relation \( R \) defined on the set \( A = \{ x : x \in \mathbb{Z}, 0 \leq x \leq 10 \} \) given by \( R = \{ (a, b) : (a - b) \text{ is divisible by } 4 \} \) is an equivalence relation, we need to verify three properties: reflexivity, symmetry, and transitivity.

### Step 1: Show Reflexivity
To show that the relation is reflexive, we need to prove that for every element \( a \) in \( A \), \( (a, a) \) is in \( R \).

- For any \( a \in A \):
\[ a - a = 0 \]
Since \( 0 \) is divisible by \( 4 \), it follows that \( (a, a) \in R \).

**Conclusion**: The relation is reflexive.

### Step 2: Show Symmetry
To show that the relation is symmetric, we need to prove that if \( (a, b) \in R \), then \( (b, a) \in R \).

- Assume \( (a, b) \in R \). This means:
\[ a - b \text{ is divisible by } 4 \]
Let \( a - b = 4k \) for some integer \( k \). Then:
\[ b - a = -(a - b) = -4k = 4(-k) \]
Since \( -k \) is also an integer, \( b - a \) is divisible by \( 4 \).

**Conclusion**: The relation is symmetric.

### Step 3: Show Transitivity
To show that the relation is transitive, we need to prove that if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

- Assume \( (a, b) \in R \) and \( (b, c) \in R \). This means:
\[ a - b = 4k \quad \text{(for some integer } k\text{)} \]
\[ b - c = 4m \quad \text{(for some integer } m\text{)} \]
Adding these two equations:
\[ (a - b) + (b - c) = 4k + 4m \]
This simplifies to:
\[ a - c = 4(k + m) \]
Since \( k + m \) is an integer, \( a - c \) is divisible by \( 4 \).

**Conclusion**: The relation is transitive.

### Conclusion
Since the relation \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Step 4: Find All Elements Related to 4
Now, we need to find all elements in \( A \) that are related to \( 4 \). An element \( x \) is related to \( 4 \) if \( (x - 4) \) is divisible by \( 4 \).

- This means:
\[ x - 4 = 4k \quad \text{for some integer } k \]
Rearranging gives:
\[ x = 4 + 4k \]

Now, we will find the values of \( k \) such that \( x \) remains in the set \( A \) (i.e., \( 0 \leq x \leq 10 \)).

1. For \( k = 0 \):
\[ x = 4 + 4(0) = 4 \]
2. For \( k = -1 \):
\[ x = 4 + 4(-1) = 0 \]
3. For \( k = 1 \):
\[ x = 4 + 4(1) = 8 \]

Thus, the elements related to \( 4 \) in the set \( A \) are \( 0, 4, \) and \( 8 \).

### Final Answer
The relation \( R \) is an equivalence relation, and the elements related to \( 4 \) are \( \{0, 4, 8\} \).