Question

8. Is the following statement true or false? Justify your conclusion. If a and b are nonnegative real numbers and a + b =0, then a = 0. Either give a counterexample to show that it is false or outline a proof by completing a know-show table.

Answer 1

Answer:

If $a + b = 0$ then $a = 0$ and $b =0$

a  | b | a + b (answer)

0 | 0 | 0

0 | 1  | 1

0 | 2 | 2

1  | 0 | 1

2 | 0 | 2

1  | 1  | 2

2 | 1  | 3

 

Step-by-step explanation:

Considering the following conditions for the real numbers:

$a\geq 0\\b\geq 0$

Following the rules of these in-equations, it is possible to deduce:

$a + b \geq 0$

Then, if the proposed statement is:

$a + b = 0$

The conditions above shall comply the requirements established, but first, analyzing the statement:

If $a \geq 0$ and $b \geq 0$ then $a + b \geq a$, $a + b \geq b$ and $a + b \geq 0$.

If $a = 0$ and b a non negative real number, then $a + b \geq b$, but because to $a = 0$, then $a + b = b$. Due to the commutative property of sums, the same behavior will be presented if $b = 0$ and a a non negative real number.

According to that, if  $a + b = 0$, then $a = 0$ and $b = 0$.