Question

Let A be the set of all triangles in a plane. Show that the relation $R=\left\{ \left( {{\Delta }_{1}},{{\Delta }_{2}} \right):{{\Delta }_{1}}\sim {{\Delta }_{2}} \right\}$ is an equivalence relation on A.

Answer 1

Hint: In order to solve this question, we will prove that the given relation R is reflexive, symmetric, and transitive because an equivalent relation satisfies all these relations. Also, we have to remember that 2 triangles are said to be similar if their corresponding angles are congruent and corresponding sides are proportional.

Complete step-by-step answer:
In this question, we have been given a relation R, that is, $R=\left\{ \left( {{\Delta }_{1}},{{\Delta }_{2}} \right):{{\Delta }_{1}}\sim {{\Delta }_{2}} \right\}$ and we have been asked to prove this as an equivalent relation on set A. So, to prove this, we will prove that the given relation R is reflexive, symmetric, and transitive relation because we know that any relation is equivalence relation only when it is reflexive, symmetric and transitive relation.
Now, let us go with reflexive relation. We know that reflexive relation is a relation which maps for itself. So, for ${{\Delta }_{1}}R{{\Delta }_{2}}$, we will see whether the triangles are similar or not. So, for that, we can say that any triangle that has all the angles congruent to the corresponding angles and all the corresponding sides are in proportion. So, we can say any triangle is always similar to itself. Hence, $R=\left\{ \left( {{\Delta }_{1}},{{\Delta }_{2}} \right):{{\Delta }_{1}}\sim {{\Delta }_{2}} \right\}$.
Therefore, relation R holds reflexive relation.
Now we will consider symmetric relations. Symmetric relation is a relation which also satisfies the converse relation. For example, if $aRb$ is satisfied, then $bRa$ should also be satisfied for symmetric relation. Here we have been given relation R, that is, $R=\left\{ \left( {{\Delta }_{1}},{{\Delta }_{2}} \right):{{\Delta }_{1}}\sim {{\Delta }_{2}} \right\}$. So, if it is true, then we have to prove that, $\left\{ {{\Delta }_{2}},{{\Delta }_{1}} \right\}$ also satisfies relation R. For that, we will start from ${{\Delta }_{1}}\sim {{\Delta }_{2}}$. We know that similarity of triangles shows commutative property, that is, ${{\Delta }_{1}}\sim {{\Delta }_{2}}$ is same as ${{\Delta }_{2}}\sim {{\Delta }_{1}}$, which is the condition for $\left\{ {{\Delta }_{2}},{{\Delta }_{1}} \right\}$. Hence, we can say that relation R is true for$R=\left\{ \left( {{\Delta }_{1}},{{\Delta }_{2}} \right):{{\Delta }_{1}}\sim {{\Delta }_{2}} \right\}$.
Therefore, we can say that relation R is a symmetric relation.
Now, let us go with a transitive relation, which states that if $aRb$ and $bRc$, then $aRc$. So, to check whether R is transitive or not, we will consider ${{\Delta }_{1}},{{\Delta }_{2}}$ and ${{\Delta }_{3}}$. And we will see that ${{\Delta }_{1}}R{{\Delta }_{2}}\Leftrightarrow {{\Delta }_{1}}\sim {{\Delta }_{2}}$ and ${{\Delta }_{2}}R{{\Delta }_{3}}\Leftrightarrow {{\Delta }_{2}}\sim {{\Delta }_{3}}$ and now we will see whether relation ${{\Delta }_{1}}R{{\Delta }_{3}}$ satisfies or not. We know that ${{\Delta }_{1}}\sim {{\Delta }_{2}}$ and we can write it as,
${{\Delta }_{2}}\sim {{\Delta }_{1}}\ldots \ldots \ldots \left( i \right)$
And also, we know that, ${{\Delta }_{2}}\sim {{\Delta }_{3}}\ldots \ldots \ldots \left( ii \right)$
So, from equation (i) and (ii), we can say that${{\Delta }_{1}},{{\Delta }_{3}}$ are similar to one triangle, that is, ${{\Delta }_{2}}$. Therefore we can write ${{\Delta }_{1}}\sim {{\Delta }_{3}}$, which is the condition of ${{\Delta }_{1}}R{{\Delta }_{3}}$.Hence, we can say R is transitive relation.
Therefore, we have proved that the given relation R is reflexive, symmetric, and transitive relation. Hence, R is an equivalent relation.

Note: While solving this question, we need to remember that a triangle is always similar to itself. Also, we should know that if ${{\Delta }_{1}}\sim {{\Delta }_{2}}$, then ${{\Delta }_{2}}$ will be definitely similar to ${{\Delta }_{1}}$. Also, to prove this question, we need to remember that the equivalent relation is the one that is a reflexive, symmetric and transitive relation.