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19

This is "left involution". ("left" because it doesn't work when you try it on the right.) \begin{align*} x \circ y &= z & \\ x \circ (x\circ y) &= x \circ z & [\text{apply $x \circ -$}] \\ y &= x \circ z & [\text{simplify the involution}] \text{.} \end{align*} I would be shocked to see anyone use that term ...

5

A helpful way to rewrite that statement would be (assuming subtraction for simplicity): $x - y - z ⇔ x - z - y$ We are observing how swapping y and z does not change the value of the expression. While it may initially look like there is a useful property behind this, the example is showing an easy case of what you are allowed to swap. Here is a visual ...

1

I don't know if this word is used specifically to describe this phenomenon, but the term "complement" is used in general to refer to two things that combine to make some third thing, so this applies here. When we subtract $b$ from $a$, we're basically asking what $b$'s (additive) complement with respect to $a$ is. Another term that could be ...

11

I have never seen a name for this property specifically. When I was in grade school, I recall learning about Fact Families, which are generated by this property. The idea is that a fact family is all of the arithmetic equations generated by the same numbers. This property in particular is really just a consequence that subtraction is the inverse of addition ...

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