Given the set $A = [1,2,3,4]$.
I am looking for examples of relations $R$ that are
- Reflexive but not symmetric, not transitive, and not anti-symmetric
- Reflexive and transitive, but not symmetric, and not anti-symmetric
I have a feeling that because of the reflexivity condition it's impossible to build relations that are both not symmetric and not anti-symmetric, but I am having a hard time showing this to be the case, any help would be appreciated. Counter-examples and/or proofs are welcome.
Best Answer
A relation that is reflexive is not symmetric if there is a counterexample to symmetry. $$\exists x\exists y: x\neq y\land xRy\land\lnot yRx$$
A relation is not anti-symmetric if there is a counterexample to anti-symmetry. $$\exists x\exists y: x\neq y\land xRy\land yRx$$
These statements may both be satisfied. Consider the following. $$\begin{bmatrix}(1,1) & (1,2) & (1,3)\\ & (2,2)\\(3,1) &(3,2)& (3,3) \\ &&& (4,4)\end{bmatrix}$$
This counterexample is, however, transitive.