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How can the type of the following ternary relation $R$ on $\mathbb{Z}$ (set of all integers) be determined whether it is reflexive, transitive or symmetric ?

$$ R = \{ a, b, c \in \mathbb{Z} : a \cdot b \cdot c < -1 \} $$

If the aforementioned properties are not applicable for a ternary relation, unlike binary relation, then what are the properties of a ternary relation?

I tried searching about the concept and the problem over the internet, but failed to get any.

A detailed answer would be helpful.

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    $\begingroup$ How do you define reflexive, transitive, symmetric for a ternary relation? $\endgroup$ – Jair Taylor May 18 at 1:38
  • $\begingroup$ Yes that is I want to know. If these properties are not applicable for a terenary relation, unlike binary relation, then what are the properties of a terenary relation ? $\endgroup$ – user 493905 May 18 at 1:40
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    $\begingroup$ Googling tells me there is such a thing as a ternary equivalence relation and definitions for these properties are given in that context. $\endgroup$ – Jair Taylor May 18 at 2:10
  • $\begingroup$ Can i please be referred to few example questions with solutions on ternary and N-nary equivalence relations ? $\endgroup$ – user 493905 May 23 at 18:23
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According to this Wikipedia page

the notions are defined as follows (I'll use $R(a,b,c)$ for $(a,b,c) \in R$), together they're called a ternary equivalence relation:

  • Symmetry: $\forall a,b,c: R(a,b,c) \implies (R(b,c,a) \land R(c,b,a))$ So all permutations of arguments also hold; th elatter I would see as the most general form of the definition: for any permutation $\sigma \in S_3$ we have $R(x_1,x_2,x_3)$ implies $R(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)})$
  • Reflexivity: $\forall a,b: R(a,b,b)$ and this implies (if we have symmetry), the most general form of the definition if $|\{a,b,c\}| \neq 3$ we have that $R(a,b,c)$ holds.
  • Transitivity: if $a \neq b$ and $R(a,b,c)$ and $R(a,b,d)$ then also $R(b,c,d)$.

The symmetry of your $R$ is evident, while $R(1,1,1)$ does not hold so reflexivity is out. I think transivity might fail too, look for examples..

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  • $\begingroup$ Can i please be referred to few example questions with solutions on ternary and N-nary equivalence relations ? $\endgroup$ – user 493905 May 23 at 18:22
  • $\begingroup$ @Aditya I presume you have a text book or notes? I just used Wikipedia... $\endgroup$ – Henno Brandsma May 23 at 21:23
  • $\begingroup$ Do you know of any online practice material or notes for this ? $\endgroup$ – user 493905 May 24 at 11:00

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