- Ex.35 from Supplemental notes
- Prove Theorem 1.1

*(it is really trivial - les than half a line per item in most cases)* - 1.6
- Prove that for any ideal
*I*an integer*a*is in*I*iff (if and only if)*aZ*is a subset of*I*. - For any integers
*a*, show that the ideal_{1},..., a_{k}*a*generated by these integers is the "smallest" ideal containing_{1}Z+...+ a_{k}Z*a*. That is for any ideal_{1},..., a_{k}*I*, prove that*a*are in_{1},..., a_{k}*I*iff the ideal*a*is a subset of_{1}Z+...+ a_{k}Z*I*.

- Prove that equality modulo
*n*is an equivalence relation on the set of integers.

- Which one is the smallest equivalence relation on integers? What are the
classes of this relation?

- Among two binary numbers of length
*n*,*n>0*, we define this relation ~ :*a~b*if and only if*a*and*b*have the same parity. Is this an equivalence relation? If yes, describe its classes.

- Let
*S*be the set of all students registered for CS235. If*x*is a student then we define course(*x*)=#of courses that*x*is taking.- What is the domain and range of this function? Write its type in the "
*f: A--> B*" - What kind of a function is this?
- If we define relation on
*S*to be*a ^ b*iff course(*a*) ≤ course(*b*), which properties does this relation have?

- What is the domain and range of this function? Write its type in the "