A reflexive functions refers back to itself and has a mathematical definition of $$∀a∈A: (a,a)∈R$$. An irreflexive function can refer to any number of elements and has the mathematical defintion of $$∀a∈A: (a,a)∉R$$ However how would I show that a relation is neither irreflexive nor reflexive?
Sorry I'm studying this in German, so their might be some translation mistakes of sorts.
First, you should note that reflexive and irreflexive aren't negations of each other; a relation can have one of the two properties, or neither of the properties, though it can't have both (unless you allow relations on the empty set). So you really have to prove two separate statements.
Assuming you're familiar with logical negation, then the negation of being reflexive is $\exists a\in A:(a,a)\notin R$ and the negation of being irreflexive is $\exists a\in A: (a,a)\in R$. These are what you have to prove to show not being reflexive and not being irreflexive respectively.