Here's the first one:
Here's an example so you can believe it:
Find this sum:
Now, find these sums and add them:
Do you believe it? (You should have gotten the same thing for both.)
This may not seem like a big deal, but you'll need it to prove some important calculus stuff, so let's get used to it.
Here's the proof:
|Just expand it out...|
Now, use the commutative and associative properties... Gather up the a's... Gather up the b's...
Here are the other two properties -- I'll let you do the proofs: