Question

If ab belongs to R if a+b=0 then a=-b???

Answer 1

Answer to Question #82334, Math / Real Analysis

Question

For $a, b$ belongs to $R$ if $a + b = 0$ then $a = -b$??

Answer

Yes, for $a, b \in R$ if $a + b = 0$ then $a = -b$

Solution

Given: $a, b \in R$

We have to prove that if $a + b = 0$ then $a = -b$

We need to prove that for every $b \in R$ there is only one additive inverse of $b$. That is if $a \in R$ with $a + b = b + a = 0$ then $a = -b$

Suppose $a$ and $a'$ are both additive inverse of $b$. Then,

$$\begin{array}{l} a = a + 0 \quad \text{by additive identity} \\ = a + (b + a') \quad \text{as } a' \text{ is an additive inverse of } b \\ = (a + b) + a' \quad \text{by associativity of addition} \\ = 0 + a' \quad \text{as } a \text{ is an additive inverse of } b \\ = a' \quad \text{by additive identity} \\ \end{array}$$

Hence, proved