Justify that if x plus y equals x plus z then y equals z?

Answer 1

The requirements are that the operation of addition is associative, the existence of an additive identity and additive inverses.

Associativity: a + (b + c) = (a + b) + c = a + b + c

Identity: The set contains a unique element, called the additive identity and often denoted by 0 with the property that a + 0 = 0 + a = a for all a in the set.

Inverse: For each element in the set, x, there exists an element which is its additive inverse. For addition, this is denoted by -x, and has the property that -x + x = 0.

x + y = x + z

Add the inverse of x to both sides: -x + (x + y) = -x + (x + z)

By associativity: (-x + x) + y = (-x + x) + z

-x is additive inverse of x, so: 0 + y = 0 + z

0 is additive identity, so y = z