I was wondering whether the Cancellation Law of Addition is the same as the Addition Property of Equality.
The Cancellation law of addition states that a= b if and only if a+c = b+c which is similar to the Addition Property of Equality that says that one can add the same quantity to both sides of an equation so if you have a = b then you can add c to both sides to get that a+c = b+c.
So is the Cancellation Law of Addition the same as the Addition Property of Equality? Do they have any differences like a proof where you can only use the Cancellation Law of Addition but and not the Addition Property of Equality?
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$\begingroup$This is one of those situations where compact notation will make the difference between the two statements clearer. Let $\implies$ stand for "implies" and let $\iff$ stand for "if and only if". By your description, here are the two laws:
- Addition Property of Equality: $a=b\implies a+c=b+c$
- Cancellation Law of Addition: $a=b\iff a+c=b+c$
Just visually speaking, in terms of the notation, what's the difference? The Cancellation Law has an arrow that goes both ways. That means the Cancellation Law is a stronger statement. It says more; it is more useful. There are more situations where you can use it.
Imagine that you have unknown quantities $x$ and $y$ and you know that $x+5=y+5$. Imagine, further, that as a small child you accidentally desecrated an ancient tomb and were cursed with a deadly allergy to negative numbers. Using the Cancellation Law, you can safely conclude that $x=y$. By contrast, you can't use the Addition Property to reach the same conclusion.
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