N is the set of natural numbers. The relation R is defined on \( \Large N \times N \) as follows: \( \Large \left(a,\ b\right)R \left(c,\ d\right) \Leftrightarrow a+d=b+c \) is:


A) reflexive

B) symmetric

C) transitive

D) all of these

Correct Answer:
D) all of these

Description for Correct answer:
(a) \( \Large \left(a,\ b\right) R \left(a,\ b\right) \Leftrightarrow \ a+b=b+a \)

Therefore, R is reflexive

(b) \( \Large \left(a,\ b\right) R \left(c,\ d\right) => a+d = b+c \)

=> \( \Large c+b = d+a \)

=> \( \Large \left(c,\ d\right) R \left(a,\ b\right) \)

Therefore, R is symmetric

(c) \( \Large \left(a,\ b \right) R \left(c,\ d\right)\ and\ \left(c,\ d\right) R \left(e,\ f \right) \)

=> \( \Large a+d=b+c\ and \ c + f=d+e \)

=> \( \Large a+b+c+f = b+c+d+e \)

=> \( \Large a+f = b+e \)

=> \( \Large \left(a,\ b\right) R \left(e,\ f\right) \)

Therefore, R is transitive

Thus R is an equivalence relation \( \Large N \times N \)

Part of solved Set theory questions and answers : >> Elementary Mathematics >> Set theory








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