Question

How to prove these ?

If `a,b ϵ R` then prove that if `a+b=0` then `b=-a` ?

Answer 1

Add `-a` to both members of the equality:

`-a+ (a+b) = -a`

using the commutative property of the sum of two real numbers:

`-a + (a+b) = (-a+a) + b`

by the definition of opposite of a real number:

`(-a+a) + b= 0 + b`

and as zero is the identity element of the sum:

`0+b = b`

Then by the transitive property of the equality of two real numbers:

`b = -a`