How to prove $\sum_{i\in I\setminus J} a_i=\sum_{i\in I} a_i-\sum_{i\in I\cap J} a_i$?

Question

Let $I,J$ be finite subsets of $\Bbb N$ and $(a_i):\Bbb N\to \Bbb R$

I define $\sum_{i\in I} a_i:=\sum_{k=1}^N a\_i_k$ where $N=|I|$ and $(i_k):\{1,....N\}\to I$ is strictly increasing and onto $I$.

Are the following propositions right? and how can I prove them:

1. "$\sum_{i\in I\setminus J} a_i=\sum_{i\in I} a_i-\sum_{i\in I\cap J} a_i$"

2. "$\sum_{i\in I\cup J} a_i=\sum_{i\in I} a_i+\sum_{i\in J\setminus I} a_i$"

3. "$\sum_{i\in I} a_i=\sum_{i\in I\cap J} a_i+\sum_{i\in I\setminus J} a_i$"

Answer 1

(1) and (3) are clearly equivalent. But actually the three assertions follow from this property:

if $E$ and $F$ are disjoint subsets of some set $U$, then $\sum_{i \in E \cup F} a_i = \sum_{i\in E} a_i + \sum_{i \in F} a_i$.