# Question regarding ternary relation

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.

• How do you define reflexive, transitive, symmetric for a ternary relation? – Jair Taylor May 18 at 1:38
• 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 ? – user 493905 May 18 at 1:40
• Googling tells me there is such a thing as a ternary equivalence relation and definitions for these properties are given in that context. – Jair Taylor May 18 at 2:10
• Can i please be referred to few example questions with solutions on ternary and N-nary equivalence relations ? – user 493905 May 23 at 18:23

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..