Proof that a linear code union with another linear code is a linear code

Question

1. Let C be a binary linear code: Proof or disproof $C∪(C+x) $ is a linear code.

For this task my solution idea is:

$c_1, c_2 \in C $ therefore (because $C$ is linear) $c_1 + c_2 \in C$

($c_1 + x) + (c_2 + x) = c_1 + c_2 +2x = c_1+c_2$

Because $C$ is linear code and all calculations are modulo 2 we do not need to show scalar multiplication. (Is this right?)

Therefore $C∪(C+x)$ is linear.

2. Let $C, D$ be binary linear codes. Proof or disproof that $C \cup D$ is also linear code.

For this task I have no idea (does task 1. help?), any help is welcome.

Answer 1

The first part is right, and there is another question that asks the same, "answered" in the comments.

The second part is false. Consider $C=\langle (1,0)\rangle$ and $D=\langle (0,1)\rangle$. It is clear that $(1,0)+(0,1)=(1,1)\notin C\cup D$, so it is not closed under addition.