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Let $Q = \{A,B,C\}$.

I argue that a relation on A that is symmetric and Anti-symmetric but neither reflexive nor transitive is the relation $R$ such that $R =\{\}$. This I think is correct because both symmetry and anti-symmetry are "if...then" statements and if the "if" is false, then it is vacuously true meaning that $\{\}$ is symmetric and anti-symmetric

I also have another answer that goes as $R = \{(A,A), (B,B)\}$. This second answer is because I interpreted neither.. nor as not ($A$ or $B$) which is not $A$ and not $B$. So, I only had to make sure it is not reflexive to make the second part false (since symmetry and anti-symmetry imply transitivity)

Which of my answers are correct please. If none, can anyone give a correct answer

River Uzoma
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  • What do you mean by a relation "that follows the exact terms above"? I don't see any terms that define a relation...I just see a set with three elements. – lulu Dec 09 '23 at 19:33
  • @lulu I meant it in reference to the question but you are right, that was unclear. So I edited it – River Uzoma Dec 09 '23 at 19:38
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    Got it. Your proposed relations are transitive, somewhat vacuously. That is to say, there are no counterexamples to transitivity in the empty relation, for example. – lulu Dec 09 '23 at 19:49
  • @lulu I see. So any example I give has to counter both of them? I thought that I could counter one. That means I can't make a relation that counters both which means such a relation does not exist which then means, I have to prove it false by proving its negation. Is this correct? – River Uzoma Dec 09 '23 at 20:02
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    Yes, I suggest trying to argue that the relation does not exist. Indeed, In order to not be transitive there needs to be some non-trivial similarity $X\sim Y$ with $X\neq Y$. And then symmetry requires that $Y\sim X$ but anti-symmetry forbids that. – lulu Dec 09 '23 at 20:03
  • @lulu Also, I wrote the negation as If a relation on S is symmetric and antisymmetric, then it is transitive or reflexive. Is that correct? – River Uzoma Dec 09 '23 at 20:04
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    Well...I'd say, more simply, that "symmetric plus anti-symmetric implies transitive." I don't think reflexivity enters into it. – lulu Dec 09 '23 at 20:05
  • @lulu Okay. That makes sense. Thank you very much – River Uzoma Dec 09 '23 at 20:07

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