I want to know if different scenarios in relations must satisfy all the value in the relation.
In mathematical relations, a given set relation is reflexive if all the elements in the set exhibit (a,a).
What about the rest of the relations such as symmetric, transitive, anti-symmetric, and so on? Can the properties be satisfied once, or does it need to be valid for any given values in a relation?
Most of the conditions are universal, so they must be true for all values. Note that the set where things are coming from does make a difference in how the properties behave. Let $R$ be a relation on $A$ (so $R\subseteq A\times A$).
Reflexive: $$ \forall a\in A, (a,a)\in R $$
Irreflexive: $$ \forall a\in A, (a,a)\not\in R $$
Symmetric: $$ \forall (a,b)\in R, (b,a)\in R $$
Antisymmetric: $$ \forall (a,b)\in R\cap R^{-1}, a=b $$ (I've written this in an atypical way, usually, it's written ``If $(a,b)$ and $(b,a)$ are in $R$, then $a=b$.)
Transitive: $$ \forall (a,b),(b,c)\in R, (a,c)\in R $$ (Transitivity is often written as an if-then as well, ``if $(a,b)$ and $(b,c)$ are in $R$, then $(a,c)$ is in $R$.)