I have looked through the web for this answer and what I get is a similar definition for both of them. This is what I found:
Monotone constraints possess the following property. If an itemset S violates a monotone constraint C, then any of its subsets also violates C. Equivalently, all supersets of an itemset satisfying a monotone constraint C also satisfy C (i.e., C is upward closed). Monotone
Anti-monotone constraints possess the following nice property. If an itemset S satisfies an anti-monotone constraint C, then all of its subsets also satisfy C (i.e., C is downward closed). Equivalently, any superset of an itemset violating an anti-monotone constraint C also violates C. Anti-monotone
Quoting the definitions in the links:
A constraint C is monotone if and only if for all itemsets S and S′: if S ⊇ S′ and S violates C, then S′ violates C.
A constraint C is anti-monotone if and only if for all itemsets S and S′: if S ⊇ S′and S satisfies C, then S′ satisfies C.
Can anyone please explain it to me what's the difference between them? Preferably with an example.
With the first definition, if S violates C, a superset of it might not also violate C but a subset must... If S satisfies C, a superset of it must also satisfy C but a subset might not.
The second definition is the other way around... if S violates C, a superset of it must also violate C but a subset might not... and if S satisfies C, a superset of it might not also satisfy C but a subset must.