Partition of Integers with mod

[nicnicman]

Homework Statement

Are the following subsets partitions of the set of integers?

The set of integers divisible by 4, the set of integers equivalent to 1 mod 4, 2 mod 4, and 3 mod 4.

Homework Equations

The Attempt at a Solution

Yes, it is a partition of the set of integers. Consider 4/4 = 1, 5/4 = 1 R 1, 6/4 = 1 R 2, 7/4 = 1 R 3.

However, how would you create a negative number like -5?

 

[Dick]

Homework Statement

Are the following subsets partitions of the set of integers?

The set of integers divisible by 4, the set of integers equivalent to 1 mod 4, 2 mod 4, and 3 mod 4.

Homework Equations

The Attempt at a Solution

Yes, it is a partition of the set of integers. Consider 4/4 = 1, 5/4 = 1 R 1, 6/4 = 1 R 2, 7/4 = 1 R 3.

However, how would you create a negative number like -5?

I'm not too clear on what your argument is supposed to mean. What's R? But -5=3 mod 4 since (-5)=(-2)*4+3. Hmm, I think I see. R means 'remainder', yes?

 

[nicnicman]

Sorry, I forgot to mention this is far from a formal proof. R just means remainder.

 

[Dick]

Sorry, I forgot to mention this is far from a formal proof. R just means remainder.

Yeah, it is far from formal. But I see what you are doing. Are you supposed to give something formal or just answer yes or no?

 

[nicnicman]

Just answer yes or no. And, you answered my question. I'm pretty sure the answer is yes. Thanks.