Which of the following relations is an equivalence relation?

An equivalence relation on a set must be Reflexive, Symmetric, and Transitive.

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Testing Option 3: $R_3 = \{(x, y) : x - y = 0\}$

The condition $x - y = 0$ can be rewritten as $x = y$.

Reflexive Property

A relation $R$ is reflexive if $(x, x) \in R$ for every real number $x$.

Check: Does $x - x = 0$?

Yes, $0 = 0$ is always true.

$\therefore R_3$ is Reflexive.

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Symmetric Property

A relation $R$ is symmetric if $(x, y) \in R \implies (y, x) \in R$.

Check: If $x - y = 0$, does $y - x = 0$?

$x - y = 0 \implies x = y$

If $x = y$, then clearly $y = x$, which means $y - x = 0$.

$\therefore R_3$ is Symmetric.

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Transitive Property

A relation $R$ is transitive if $(x, y) \in R$ and $(y, z) \in R \implies (x, z) \in R$.

Check: If $x - y = 0$ and $y - z = 0$, is $x - z = 0$?

$x - y = 0 \implies x = y$

$y - z = 0 \implies y = z$

From the above, $x = z$. Therefore, $x - z = 0$.

$\therefore R_3$ is Transitive.

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Why the other options are incorrect:

$R_1$ ($3x - y = 10$): Not Reflexive because $3x - x \neq 10$ for all real numbers (e.g., if $x=1$, then $3(1)-1 = 2 \neq 10$).

$R_2$ ($x - y = 10$): Not Reflexive because $x - x \neq 10$ (since $0 \neq 10$).

$R_4$ ($x - 3y = 0$): Not Reflexive because $x - 3x \neq 0$ for all real numbers (only true for $x=0$).