Let N denote the set of all natural numbers.

Question

Let N denote the set of all natural numbers. Define two binary relations on N as R = {(x, y) $$ \in $$ N $$ \times $$ N : 2x + y = 10} and R2 = {(x, y) $$ \in $$ N $$ \times $$ N  :  x  +  2y = 10}. Then :

Accepted Answer

For R1; 2x + y = 10 and x, y $$ \in $$ N possible values for x and y are :

x = 1, y = 8    i.e.   (1, 8);

x = 2, y = 6    i.e    (2, 6);

x = 3, y = 4    i.e    (3, 4);

x = 4, y = 2    i.e    (4, 2)

$$\therefore\,\,\,$$ R1 = { (1, 8), (2, 6), (3, 4), (4, 2) }

$$\therefore\,\,\,$$ Range of R1 is {2, 4, 6, 8}

R1 is not symmetric.

R1 is not transitive also as

(3, 4), (4, 2) $$ \in $$ R , but (3, 2) $$ \notin $$ R1

For R2 : x + 2y = 10 and x, y $$ \in $$ N

Possible values of x, and y are :

x = 8, y= 1    i.e    (8, 1)

x = 6, y = 2    i.e    (6, 2)

x = 4, y = 3    i.e    (4, 3) and

x = 2, y = 4    i.e    (2, 4)

$$\therefore\,\,\,$$ R2 = {(8, 1) (6, 2) (4, 3) (2, 4)}

$$\therefore\,\,\,$$ Range of R2 = $$\left\{ {1,2,3,4} \right\}$$

R2 is not symmetric and transitive

Solutions · 1