Can someone elaborate on what rules underlie the rule


Is it associative properties of multiplication and addition? For example,





Are there any exceptions to $-a+b=b-a$?

  • 4
    $\begingroup$ I would say commutativity of addition and definition of subtraction as addition of the additive inverse $\endgroup$ Feb 12 '19 at 3:41

We don't have to invoke any properties of multiplication to explain why $-a+b=b-a.$ We don't have to invoke associativity (which refers to different ways of grouping three operands) either. We merely have to know that addition is commutative (i.e., $x+y=y+x$) and every element $z$ has an additive inverse ($-z$) such that $z+ (-z) = 0.\;$ We define subtraction by $x-z=x + (-z)$. In any system where those assumptions hold, it is always true that $(-a) + b = b + (-a) = b -a$.


The commutative property of addition.

$$x+y = y+x$$.

Set $x = -a$ and $y=b$.

There are no exceptions.


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