# Proving if a set is closed under addition, subtraction, and division

$M$ is the set of all multiples of $11$, i.e., $M=\{11k: k\in \Bbb Z\}$.

a. Is $M$ closed under addition?

b. Is $M$ closed under subtraction?

c. Is $M$ closed under division?

So, I used the multiples of $11$ and started performing the operations stated above. I am finding that it is closed under addition, subtraction, and sometimes division. I do not know how to show a formal proof of my findings.

• $\text{a. }\forall x,y \in M: x+y \in M?\\\text{b. }\forall x,y \in M: x-y \in M?\\\text{c. }\forall x,y \in M: \dfrac xy \in M?$
– DHMO
Commented Apr 13, 2017 at 14:54

You say "sometimes division." If you have only one counter example where the result of an operation is not in the set, then the set is not closed under that operation. Therefore, you have given the answer for (c).

Indeed $22 ÷ 11 = 2$, which is not in $M$, so the set is not closed under division.

You are correct that the set is closed under addition. A formal proof might look like this:

Let $x, y \in M$
By definition, $x=11j$ for some $j \in \mathbb{Z}$ and $y=11k$ for some $k \in \mathbb{Z}$
Then $x+y = 11j + 11k = 11(j+k) \in M$

You can do something similar for subtraction.

It is closed under operations of addition and subtraction since $k\mathbb{Z}$ is an ideal of $\mathbb{Z}.$ However, it isn't closed under division, since $\frac{11k}{11k}=1,$ and since $11\not\mid 1$ this is a suitable counter example.