The additive property of equality stated algebraically is: For real or complex numbers a, b, and c, if a = b then a + c = b + c. This means that we can add the same expression to both sides of an equation and the new equation is still has the same truth value.
The additive property of equality applies to any relation on a set which is:
|1||Let a, b and c be members of a set S and '=' be a relationship on that set.||Define the mathematical objects needed for the proof.|
|2||Let a = b.||Establish the criterion.|
|3||Let d = a + c, d ∈ S.||Define member 'd' of the set. Note that this step requires that the set S be closed with respect to addition.|
|4||Then d = b + c.||Use the substitution property of equality to substitute b for a.|
|5||d = d.||Use the reflexive property of equality to establish d = d.|
|6||a + c = d.||Use the substitution property of equality to substitute a + c in for d.|
a + c = b + c.|
|Use the substitution property of equality to substitute b + c in for d.|
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