Question

Let R be a relation on $N\times N$ defined by $(a,b)R,(c,d)\iff a+d=b+c$ for all $(a,b),(c,d)ϵN\times N$

Show that :

(i) (a, b) R (a, b) for all $(a,b)ϵN\times N$

(ii) $(a,b)R(c,d)\Rightarrow (c,d)R(a,b)$ for all $(a,b),(c,d)ϵN\times N$

(iii) (a, b) R (c, d) and (c, d) R (e, f) $\Rightarrow$ (a, b) R (e, f) for all (a, b), (c, d), (e, f) $ϵN\times N.$

Answer 1

We have,

$(a,b)R,(c,d)\iff a+d=b+c$ for all $(a,b),(c,d)ϵN\times N$

(i) We have,

a + b = b + a for all $a,bϵN$

$\therefore (a,b)R(a,b)$ for all, $a,bϵN$

(ii) Now,

(a, b) R (c, d)

$\Rightarrow a+d=b+c$

$\Rightarrow c+d=d+a$

$\Rightarrow (c,d)R(a,b)$

(iii) Now,

(a, b) R (c, d) and (c, d) R (e, f)

$\Rightarrow a+d=b+c$ and c + f = d + e

$\Rightarrow a+d+c+f=b+c+d+e$ (Adding)

$\Rightarrow a+f=b+e$

$\Rightarrow (a,b)R(e,f)$