Question

Let R be a relation defined on the set of natural numbers N as
R= {(x, y): x$\in N,y\in N,2x+y=41$}
Find the domain and range of this relation R. Also, verify whether R is (i) reflexive (ii) symmetric (iii) transitive.

Answer 1

Neither.
The relation R can be written as
R ={(1, 39), (2, 37), (3, 35), ........(10, 21).(11, 19), ..........(21, 1)}
$\therefore$ Domain of R = {1, 2, 3,..........19,20}
Range of R ={ 39, 37, 35,..........9, 7, 5, 3, 1}

For reflexive let's y=x so that 2x+x=41$\Rightarrow$ x=$\frac{41}{3}$ but

R is not reflexive as x=$\frac{41}{3}\notin N$
R is not symmetric since ( 1, 39) $\in$ R but ( 39, 1)$\notin$ R.
R is not transitive because (20, 1) $\in$ R and ( 1, 39) $\in$ R but (20, 39) $\notin$ R.