> Which of these collections of subsets are partitions > of the set of integers? > > a. the set of even and the set of odd numbers Yes, every integer is either even or odd but not both
> b. the set of positive integers and the set of > negative integers No, the integer "0" is neither postive nor odd.
> c. the set of integers divisible by 3, the set of > integers leaving a remainder 1 when divided by 3, and > the set of integers leaving a remainder of 2 when > divided by 3. Yes, every integer is of the form one of 3k, or 3k+1, or 3k+2 but never two of those.
> d. the set of integers less than -100, the set of > integers with absolute value not exceeding 100, and > the set of integers greater than 100. Yes, any integer that is not "less than -100" or "larger than 100" must be between -100 and 100 so its absolute value is less than or equal to 100.
> e. the set of integers not divisible by 3, the set of > even integers, and the set of integers that leave a > remainder of 3 when divided by 6. Yes. If a number IS divisible by 3 (so not in the first set) is a multiple of 3. If it also even, it is a multiple of 6 and so not in the third set. If it is divisible by 3 and not even, it has remainder 3 when divided by 6. (I had to reread this several times!)