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I am using "Foundations of Mathematical Economics" by Michael Carter. The problem 1.16 of the book: Consider the relations $<$, $\le$ and $=$ on $\mathbb R$. Which of the above properties (reflexive,complete,transitivity,symmetric,asymmetric,antisymmetric) do they satisfy?

About the complete properties, $<$ is NOT complete because either $A < B$ or $A > B$ BUT not both. Also, $=$ is complete because $A=B$ or $B=A$ or both, but the solution manual shows that $<$ IS complete and $=$ is NOT complete.

Is this an error? Truthfully, this book is good but full of convoluted details without mentioning in the book.

I thank you very much.

Michael Hardy
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SON TO
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If is not true that for EVERY value of $A$ and $B$ either $A=B$ or $B=A$ or both. For example, if $A=1$ and $B=2$, then this is false. If a statement says "For every . . . '", then the way to show it is false is to show that there is a counterexample.

The relation $\le$ is complete because for every $A$ and every $B$, either $A\le B$ or $B\le A$ or both. The relation $<$ is not complete because instances in which $A=B$ are counterexamples.

Possibly the authors used a definition of "complete" that said that for every $A$ and every $B$, if $A$ is not the same as $B$, then either $A$ is related to $B$ or $B$ is related to $A$ or both.

Michael Hardy
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  • I thank you very much for your answer. But what you have said "showing counterexamples" is not very tight(I think). < is complete "by vacuity". The definition of "complete": either xRy or yRx or both. Also, if EVERY VALUE OF A AND B must be satisfied, no relation will have completeness property. – SON TO May 29 '13 at 15:21
  • @SONTO : Fortunately you're mistaken: the relation $\le$ on $\mathbb R$ is indeed complete: for EVERY pair of values $A$ an $B$, it is true that either $A\le B$ or $B\le A$. And the reason why $=$ is NOT complete is that it is not true that FOR EVERY pair of numbers $A$ and $B$, either $A=B$ or $B=A$ or both. – Michael Hardy May 29 '13 at 23:17
  • The definition of "complete" that you need is that FOR EVERY pair of values $x$ and $y$, either $xRy$ or $yRx$ or both. You're missing the quantifier that says "FOR EVERY". What could it mean to say $xRy$ unless you say "FOR EVERY pair of values $x$ and $y$...." or "FOR SOME...." or the like. Completeness would then be a property of the particular pair $x,y$, rather than of the relation. For some pairs $x,y$ a relation would then be complete and for others it would not. Can you quote the textbook's definition verbatim? – Michael Hardy May 29 '13 at 23:21
  • OK, now I've looked at an excerpt via Google Books. It does indeed omit the quantifier "For every". That is sloppy. But notice that he says a relation $R$ is reflexive if $xRx$. The universally standard devinition says: For EVERY value of $x$ in the domain, $xRx$. And similarly for the others. It's symmetric if for EVERY pair of values $x,y$, if $xRy$ then $yRx$. – Michael Hardy May 29 '13 at 23:37