solve symmetric condition in another method

Certainly! Let's examine the symmetric condition of the relation \( R \) using a different approach. We need to show that if \( a \, R \, b \) (i.e., \( 2a + 3b \) is a multiple of 5), then \( b \, R \, a \) (i.e., \( 2b + 3a \) is also a multiple of 5).

### Given:
\( a \, R \, b \) means \( 2a + 3b \equiv 0 \pmod{5} \).

This can be written as:
\[
2a + 3b = 5k
\]
for some integer \( k \).

### To Prove:
We need to show that \( 2b + 3a \) is also a multiple of 5. 

Let's rewrite \( 2a + 3b = 5k \) in a different form:
\[
3b \equiv -2a \pmod{5}
\]
or equivalently:
\[
3b \equiv 3(5 - 2a) \pmod{5}
\]
which simplifies to:
\[
b \equiv -2a \times 3^{-1} \pmod{5}
\]
where \( 3^{-1} \) is the multiplicative inverse of 3 modulo 5.

#### Step 1: Find the Inverse of 3 modulo 5
We need to find an integer \( x \) such that:
\[
3x \equiv 1 \pmod{5}
\]
Testing for values:
- \( 3 \times 1 = 3 \) (not congruent to 1 modulo 5)
- \( 3 \times 2 = 6 \equiv 1 \pmod{5} \)

So, \( 3^{-1} \equiv 2 \pmod{5} \).

#### Step 2: Apply the Inverse
Substitute the inverse back:
\[
b \equiv -2a \times 2 \pmod{5}
\]
Simplifying:
\[
b \equiv -4a \equiv 1a \equiv a \pmod{5}
\]
So:
\[
b \equiv a \pmod{5}
\]
This implies that \( 2b + 3a \equiv 2a + 3b \equiv 0 \pmod{5} \), meaning \( b \, R \, a \) holds.

This method confirms that if \( a \, R \, b \) (i.e., \( 2a + 3b \) is a multiple of 5), then \( b \, R \, a \) (i.e., \( 2b + 3a \) is also a multiple of 5). Therefore, the relation \( R \) is symmetric.

Would you like more details or have any questions?

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Here are 8 related questions you might find interesting:
1. What is the significance of finding the multiplicative inverse in modular arithmetic?
2. How do you determine if a relation is symmetric without using specific examples?
3. Can modular arithmetic always be used to prove symmetry in relations?
4. What other properties can be analyzed using modular arithmetic in relation theory?
5. How does symmetry differ in relations defined over different algebraic structures?
6. What are other common methods to prove the symmetry of a relation?
7. Can symmetry fail even if reflexivity holds in a relation? Explain with examples.
8. How does the concept of congruences help in proving properties of relations?

**Tip:** Mastering the concept of modular inverses is crucial for solving various problems in number theory and cryptography. It’s a fundamental tool in modular arithmetic!

Explore a different approach to solve the symmetric condition in relations using modular arithmetic. Understand how to apply modular inverses and theorems on symmetry in relations. Suitable for advanced mathematics enthusiasts.

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