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# Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $aRb$. If a is congruent to b where $a,b\in T$, then $R$ is:

(A) reflexive but not transitive
(B) transitive but not symmetric
(C) equivalence
(D) none of these

Toolbox:
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
• .A relation R is an equivalnce relation if R is reflexive,symmetric and transitive.
Step1:
a R b a is congruant to b $a,b \in T$
$a R a$ is true since every triangle is congruent t itself
R is reflexive
Step2:
a R b
=> a is congruent to b
=> bis congruent to a
b R a
R is symmetric
Step3:
a R b and b R c
a is congruent to b and b is congruent to c
hence a is congruent to c
a R c
R is transitive
Solution:Hence R is an equivalnce relation
option 'c' is correct

edited Mar 27, 2013 by meena.p