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Many sources show this definition for completeness of a relation $$ \forall a, b \in A, a \neq b, (aRb) \text{ or } (bRa) \tag{1} $$ Others show only $$ \forall a, b \in A, (aRb) \text{ or } (bRa) \tag{2} $$

My question, assuming that the first definition is correct, why does the definition of a complete relation include the $a \neq b$ in its definition? My doubt is that when given a relation and asked whether it is complete or not, which definition should I use (1) or (2) as given in the question. And why should I use that definition? For example- Given a relation On RxR R={(x,y)| x is not equal to y}. Is this relation complete? Now if I use definition (1), it is complete. If I use definition (2), it is not complete. So which is the right answer and why?

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  • $\begingroup$ My doubt in simple words is that when given a relation and asked whether it is complete or not, which definition should I use (1) or (2) as given in the question. And why should I use that definition? For example- Given a relation On RxR R={(x,y)| x is not equal to y}. Is this relation complete? Now if I use definition (1), it is complete. If I use definition (2), it is not complete. So which is the right answer and why? $\endgroup$ – DARE2ZLATAN Nov 1 '19 at 9:20

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