Prove the following:
- if and only if .
- .
- .
- .
- .
- .
- If then .
- .
- .
- .
-
Proof. If then .
If , then by definition of -
Proof. If , then and .
If , then and (since -
Proof. If , then , so and .
If , then , so and .
If , then so -
Proof. First, . So, if then .
If then - Proof. Since is defined to be the unique non-negative square root of , from part (d) we have , and is non-negative by definition; hence,
-
Proof. If , then , so Further, , so .
If and , then so . Then, and , so
If and , then so . Then and , so
Finally, if and , then so . Then and , so -
Proof. Using the definition of we have
by part (f). But then since , we have . So,
-
Proof. Using the triangle inequality and the fact that ,
-
Proof. Here we use the trick of adding and subtracting , and then the triangle inequality,
-
Proof. Using a similar trick to part (i), we have
Then, combining this inequality with the one in part (i) and applying the definition of the absolute value,
b) |-(-5)|=-5…
|-(-5)| = -(-5)
What a wonderful site keep it up
Actually I would make a small alteration: in part (f), you should have that if and < 0, then (just to be more precise)
hmm good….
YOUR SOLUTIONS ARE WONDERFUL. YOU WILL BE A GREAT MATHEMATICIAN,.
THAT IS WHAT I SUSPECT.