A) Reflexive only
B) Symmetric only
C) Transitive only
D) An equivalence relation
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 \[(c,d)R(e,f)\] |
\[\Rightarrow (a,b)R(e,f)\] |
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