Let T be the set of all triangles in a plane with R a relation in T given by `R={(T_1,T_2): T_1( i s c o n g r u e n t t o T)_2}`. Show that R is an equivalence relation.

Options

A. reflexive but not transitive
B. transitive but not symmetric
C. equivalence
D. None of these

Answer

C

Explanation

Consider that aRb, if a is congruent to `b, AA a, b in T.`
Then, aRa `impliesa ~= a,`
which is true for all `a in T " "`…(i)
So, R is reflexive,
Let `aRb implies a ~= b`
`implies b ~=a impliesb~=a`
`implies bRa`
So, R is symmetric.
Let aRb and bRc
`implies a ~=b and b~c`
`implies a~=cimplies aRc`
So, R is transitive.
Hence, R is equivalence relation.