Question

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

Answer 1

Domain of R is the values of x and range of R is the values of y that together should satisfy 2x+y = 41.
So,
Domain of R = {1, 2, 3, 4, ... , 20}
Range of R = {1, 3, 5, ... , 37, 39}

Reflexivity: Let x be an arbitrary element of R. Then,
$$\begin{gathered} x\in R \\ \Rightarrow 2x+x=41\mathrm{cannot}\mathrm{be}\mathrm{true}. \\ \Rightarrow \left(x,x\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{reflexive}. \end{gathered}$$

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(x,y\right)\in R.\mathrm{Then}, \\ 2x+y=41 \\ \cancel{\Rightarrow }2y+x=41 \\ \Rightarrow \left(y,x\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{symmetric}. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Let}\left(x,y\right)\mathrm{and}\left(y,z\right)\in R \\ \Rightarrow 2x+y=41\mathrm{and}2y+z=41 \\ \Rightarrow 2x+z=2x+41-2y41-y-2y=41-3y \\ \Rightarrow \left(x,z\right)\notin R \\ \mathrm{Thus},R\mathrm{is}\mathrm{not}\mathrm{transitive}. \end{gathered}$$