Are These Collections of Subsets Partitions of the Set of Integers?

[brad sue]

Hi I need help with this problem I have some trouble with partitions:

Homework Statement

The context is Discrete math /relation

Which of these collections of subsets are partitions of the set of integers?

1- The set of even integer and the set of odd integers.

2- the set of positive integer and the set of negative integers.

3- the set of integers divisible by 3, the set of integers leaving a remainder of 1 when divided by 3, and the set of integers divisible by 3, the set of integers leaving a remainder of 2 when divided by 3.

4- 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.

5- the set of integers not divisible by 3, the set of even integers and the set of integer that leave a remainder of 3 when divided by 6.

Homework Equations

Are my answers correct?

The Attempt at a Solution

I found yes for all cases . but I am suspicious.

Thank you for your help
B

 

[HallsofIvy]

Hi I need help with this problem I have some trouble with partitions:

Homework Statement

The context is Discrete math /relation

Which of these collections of subsets are partitions of the set of integers?

The definition of "partition" of a set is a collection of subsets such that every member of the orginal set (here the set of all integers) is in one and only one of the subsets.

1- The set of even integer and the set of odd integers.

Is there any integer that is in neither of these sets- is there any integer that is neither even nor odd? Is there any integer that is in both- is there any integer that is both even and odd? If your answer is "yes" to either, then this is not a partition. If it is "no" to both, then it is a partition.

2- the set of positive integer and the set of negative integers.

Again, is there any integer that is neither positive nor negative? Is there any integer that is both positive and negative?

3- the set of integers divisible by 3, the set of integers leaving a remainder of 1 when divided by 3, and the set of integers divisible by 3, the set of integers leaving a remainder of 2 when divided by 3.

Is there any integer for which two of these is true- can an integer have two different remainders when divided by 3? Is there any integer for which none of these is true? What are the possible remainders when you divide a number by 3?

4- 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.

Do you see what questions you should ask yourself for this? Is there any integer for which two of these is true? Is there any integer for which none of these is true?

5- the set of integers not divisible by 3, the set of even integers and the set of integer that leave a remainder of 3 when divided by 6.

Homework Equations

Are my answers correct? ?What were your answers?

The Attempt at a Solution

I found yes for all cases . but I am suspicious.

Thank you for your help
B

Oh, I see. Your answered "yes" for every question. No, that is not true. Go back and answer the questions I asked for each one.

 

[brad sue]

Thank you HallsofIvy
I understand what I need to ask myself .
I have an exam in 15 min. I going to go now and work on it.
Thank you..