Let $T$ be the set of all triangles in a plane with $R$ as the relation in $T$ given by $R=\{(T_{1},T_{2}):T_{1} \cong T_{2}\}$. Show that $R$ is an equivalence relation.

CBSE Class 12 · 2008 · Relations And Functions XIILong Answer

Let

T

be the set of all triangles in a plane with

R

as the relation in

T

given by

R = {(T_{1}, T_{2}) : T_{1} ≅ T_{2}}

. Show that

R

is an equivalence relation.

Answer: $R$ is an equivalence relation.

We verify the three properties of congruence

≅

T

.
Reflexive: every triangle is congruent to itself,

T_{1} ≅ T_{1}

, so

(T_{1}, T_{1}) \in R

for all

T_{1} \in T

.
Symmetric: if

T_{1} ≅ T_{2}

then

T_{2} ≅ T_{1}

, so

(T_{1}, T_{2}) \in R \Rightarrow (T_{2}, T_{1}) \in R

.
Transitive: if

T_{1} ≅ T_{2}

and

T_{2} ≅ T_{3}

then

T_{1} ≅ T_{3}

, so

(T_{1}, T_{2}), (T_{2}, T_{3}) \in R \Rightarrow (T_{1}, T_{3}) \in R

.
All three hold.

R is an equivalence relation

CBSE Class 12 Board · 2008 · Subjective