Additive Property of Equality

Pronunciation: /ˈæd ɪ tɪv ˈprɒ pər ti ʌv ɪˈkwɒl ɪ ti/ ?

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:

Proof of Additive Property of Equality
StepEquationDescription
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.
7 a + c = b + c.
QED.
Use the substitution property of equality to substitute b + c in for d.

Cite this article as:


Additive Property of Equality. 2009-12-24. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/additivepropofequality.html.

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Revision History


2009-12-24: Fixed typographical error in step 7 of proof. (McAdams, David.)
2009-02-12: Added limitation for real numbers (McAdams, David.)
2008-10-05: Added proof, discussion for systems other than real numbers (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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