Question

Let $ A=\{1,2,3,4\} $ and R be a relation in A given by $ R=\{(1,1),\ \text{(2,2)},\,(3,3),\,\text{(4,4), (1,2), (2,1), (3,1), (1,3) }\!\!\}\!\!\text{ } $ then identify the relation R.
A.Reflexive
B.Symmetric
C.Transitive
D.An equivalence relation

Answer 1

Hint: An equivalence relation is the relationship on the set which is reflexive, symmetric and transitive for everything in the set. So, we check first for the reflexivity, the symmetry and the transitivity of the given set and then will decide accordingly the relation R.

Complete step-by-step answer:
Let $ A=\{1,2,3,4\} $ and R be a relation in A given by $ R=\{(1,1),\ \text{(2,2)},\,(3,3),\,\text{(4,4), (1,2), (2,1), (3,1), (1,3) }\!\!\}\!\!\text{ } $
A reflexive relation contains every ordered pair (a,a) such that $ a\in A. $ But this doesn't mean that it should not contain an ordered pair (a,b) such that $ a,b\in A. $
 $ \Rightarrow $ Now, for all $ 1,\text{ 2, 3, 4 }\in \text{ A, (1,1), (2,2), (3,3), (4,4) }\in R, $ this gives the relation R is the reflexive.
Again, for Symmetric Relation – Let A be a set in which the relation R defined, then R is said to be a symmetric relation, if $ (a,b)\in R\Rightarrow (b,a)\in \text{R} $
 $ \begin{align}
& (1,2)\ \in \text{R}\Rightarrow \text{(2,1)}\in \text{R} \\
& \text{and (1,3)}\in \text{R}\Rightarrow (3,1)\in \text{R} \\
\end{align} $
Therefore the given function is symmetric.
Now, for the transitive Relation – Let A be a set in which the relation R defined. R is said to be transitive, if $ (a,b)\in \text{R}\,and\text{ (b,a)}\in \text{R}\Rightarrow (a,c)\in R $
 $ (2,1),(1,3)\in \text{R} $ but $ (2,3)\notin R $
Therefore, the required solution is – if $ A=\{1,2,3,4\} $ and R be a relation in A given by $ R=\{(1,1),\ \text{(2,2)},\,(3,3),\,\text{(4,4), (1,2), (2,1), (3,1), (1,3) }\!\!\}\!\!\text{ } $ then the relation R is Reflexive symmetric but not transitive.
Hence, from the given multiple choices- given options are not correct.

Note: An equivalence relation is the relationship on a set, generally denoted by $ ''\sim '' $ which is reflexive, symmetric and transitive for everything in the given set. Equivalence relations are frequently used to group together objects that are similar or the equivalent in common.