Question

R :{ 1, 2, 3}$ \to $ {1,2,3} Given by $R = \{ (1,1),(2,2),(3,3),(1,2)\} $. Check if $R$ is
(a) Reflexive
(b) Symmetric

Answer 1

Hint:
To check for reflexive and symmetric we will move with the definition of reflexive and symmetric relation definition. Reflexive relation defined as if $a$ belong to the domain of relation R then (a, a) should be in relation R for every $a$ belongs to the domain of R. For symmetric relation it states that if (a, b) belong to relation R such that $a$ belongs to the domain of R and $b$ belongs to co-domain of R then (b, a) should belong to R such that b belongs to the domain of R and $a$ belong to co-domain of R.

Complete step by step solution:
Domain of Relation R =$\{ 1,2,3\} $ and Co-domain of relation R $ = \{ 1,2,3\} $ and given: $R = \{ (1,1),(2,2),(3,3),(1,2)\} $

For Reflexive: $(a,a) \in R,\forall a \in $ Domain of $R$ Here, in $R,\;\;\;{\kern 1pt} \{ (1,1)(2,2)(3,3)\} $ all are present means for $1 \in 1,2,3$ there is $(1,1) \in R$. Similarly (2,2) and (3,3) belongs to R. Hence, it satisfies the reflexive relation.
Therefore, R is a reflexive relation.

For Symmetric: if $(a,b) \in R$ such that $a \in $ Domain of R and $b \in $ Co -domain of R then $\left( {b,a} \right) \in R$ where $b \in $ Domain of R, and a $ \in $ co-domain in $R \cdot $
  (1,1)$ \to (1,1)$ [True]
  (2,2)$ \to (2,2)$ [True]
 (3,3)$ \to (3,3)$ [True]
but (1,2) does not have (2,1) in R .Hence, it is not Symmetric

Given $R$ is Reflexive but not symmetric.

Note:
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in the relation. A function is a type of relation.