Prove that an additive inverse is unique?

To prove that an additive inverse is unique, we need to show that for any element a in a set with an addition operation, there exists a unique element b such that a + b = 0.

Let's assume that there are two additive inverses for a given element a, let's call them b and c. This means that a + b = 0 and a + c = 0.

Now, let's consider the equation a + b = a + c. We can subtract a from both sides of the equation to get b = c.

Therefore, we have shown that if there are two additive inverses for a given element a, they must be equal. This proves that the additive inverse is unique.