Properties of Operations, math homework help

1. Determine which of the operations on the positive integers are: commutative; associative; idempotent; which have identities; and of those, which have inverses for all elements.

(a) addition

(b) multiplication

(c) M defined by aM b = larger ofa andb

(d) m defined by amb = smaller of a and b

(e) @ defined by a@b = a^b

2. Prove that intersection of closed subsets is closed.