Addition Property of Equality
Addition Property of Equality
An operation works to change numbers. (The word operate comes from Latin operari, "to work.")
There are six operations in arithmetic that "work on" numbers: addition, subtraction, multiplication,
division, raising to powers, and taking roots. A binary operation requires two numbers. Addition is a
binary operation, because "5 +" doesn't mean anything by itself. Multiplication is another binary
operation.
Equality
The equals sign in an equation is like a scale: both sides, left and right, must be the same in order for
the scale to stay in balance and the equation to be true. The addition property of equality says that if a =
b, then a + c = b + c: if you add the same number to (or subtract the same number from) both sides of
an equation, the equation continues to be true. The multiplication property of equality says that if a = b,
then a * c = b * c: if you multiply (or divide) by the same number on both sides of an equation, the
equation continues to be true. The reflexive property of equality just says that a = a: anything is
congruent to itself: the equals sign is like a mirror, and the image it "reflects" is the same as the original.
The symmetric property of equality says that if a = b, then b = a. The transitive property of equality
says that if a = b and b = c, then a = c.
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Equality is the state of being quantitatively the same. More formally, equality (or the identity relation)
is the binary relation on a set X defined by . The identity relation is the archetype of the more general
concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and
transitive. The relation of equality is also antisymmetric.
These four properties uniquely determine the equality relation on any set S and render equality the only
relation on S that is both an equivalence relation and a partial order. It follows from this that equality is
the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence
relation on S. An equation is simply an assertion that two expressions are related by equality (are
equal). The equality relation is always defined such that things that are equal have all and only the same
properties. Some people define equality as congruence.
Often equality is just defined as identity. A stronger sense of equality is obtained if some form of
Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"--things that
have all and only the same properties but are not equal to each other--which are possible in some
logical formalisms. The axiom states that two things are equal if they have all and only the same
properties. Formally
Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).
In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the
original. Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of
equality. The property of being an equivalence relation, as well as the properties given below, can then
be proved: they become theorems. If a=b, then a can replace b and b can replace a.
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