CS-235. Problem Set 2
- Ex.35 from Supplemental notes
- Prove Theorem 1.1
(it is really trivial - les than half a line per item in most cases)
- 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 a1,..., ak , show that the
ideal a1Z+...+ akZ generated by these
integers is the "smallest" ideal containing a1,..., ak.
That is for any ideal I, prove that a1,..., ak
are in I iff the ideal a1Z+...+ akZ is a
subset of I.
- Prove that equality modulo n is an equivalence relation on the set
- 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" format.
- What kind of a function is this?
- If we define relation on S to be a ^ b iff
course(b), which properties does this relation have?