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.
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.