Is antireflexive and irreflexive relations the same thing? Also, symmetric and antisymmetric relations?antireflexive
Yes: antireflexive means the same thing as irreflexive.
I don’t get your second question. A symmetric relation is one where R(a,b) ⇒ R(b,a) for all a, b. An antisymmetric relation is one where R(a,b) ⇒ R(b,a) only holds true if a = b. An aymmetric relation goes a step further and says that R(a,b) ⇒ R(b,a) never holds – not even if a = b. This means that aymmetric relations are necessarily irreflexive.