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Ben Brink
Posts:
201
From:
Rosenberg, TX
Registered:
11/11/06


RE: Discrete Math
Posted:
May 3, 2012 10:01 PM



Great problem and a great set of proofs! Thanks, Ben
> Date: Thu, 3 May 2012 13:18:21 0400 > From: discussions@mathforum.org > To: discretemath@mathforum.org > Subject: Re: Discrete Math > > > 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!)



