Properties of Equality

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The properties of equality are there as a set of guidelines that can be followed to solve equations in Algebra. They are the models by which we can modify an equation and still maintain it's equality.

Equality is the paradigmatic example of the more general concept of equivalence relations 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.

The symbol "=" is sometimes used for relations other than equality. For example, the statement T(n) = O(n²) means that T(n) grows at the order of n². Despite the notation, the statement is better understood as asserting a set membership: O(f(n)) is formally the set of all functions on the positive integers that, for large n, grow no faster than f(n). In particular, since membership, unlike equality, is not symmetric, it is meaningless to write O(n²) = T(n).

Some basic logical properties of equality

The substitution property states:

• For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b).

Some specific examples of this are:

• Addition Property - For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);
• Subtraction Property - For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);
• For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);
• For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.

The symmetric property states:

• For any quantities a and b, if a = b, then b = a.

The transitive property states:

• For any quantities a, b, and c, if a = b and b = c, then a = c.