Functions Question 469
Question: Let R be a relation over the $ N\times N $ and it is defined by (a, b) R (c, d)
$ \Rightarrow a+d=b+c. $ Then R is
Options:
A) Reflexive only
B) Symmetric only
C) Transitive only
D) An equivalence relation
Show Answer
Answer:
Correct Answer: D
Solution:
[d] we have $ (a,b)R(a,b) $ for all $ (a,b)\in N\times N $ As $ a+b=b+a. $ Hence, R is reflexive R is symmetric for we have (a, b) R (c, d)
$ \Rightarrow a+d=b+c $
$ \Rightarrow d+a=c+b $
$ \Rightarrow a+b=b+c $
$ \Rightarrow (c,d)R(e,f) $
Then, by definition of R, we have
$ a+d=b+c $ and $ c+f=d+e $
So, by addition, we get
$ a+d+c+f=b+c+d+c $ 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) $
BETA
