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.
Best Answer
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:
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..