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1.Relation and Function
hard
Let $R$ be a relation over the set $N × N$ and it is defined by $(a,\,b)R(c,\,d) \Rightarrow a + d = b + c.$ Then $R$ is
A
Reflexive only
B
Symmetric only
C
Transitive only
D
An equivalence relation
Solution
(d) We have $(a,b)R(a,b)$ for all $(a,b) \in N \times N$
Since $a + b = b + a$. Hence, $R$ is reflexive.
$R$ is symmetric for we have $(a,b)R(c,d)$ ==> $a + d = b + c$
==> $d + a = c + b$ ==> $c + b = d + a \Rightarrow (c,d)R(e,f).$
Then by definition of $R$, we have
$a + d = b + c$ and $c + f = d + e$,
whence by addition, we get
$a + d + c + f = b + c + d + e$ or $a + f = b + e$
Hence,$(a,b)\,\,R\,(e,f)$
Thus, $(a, b)$ $R(c,d)$ and $(c,d)R(e,f) \Rightarrow (a,b)R\,(e,\,\,f)$.
Standard 12
Mathematics