The transitive property of equality is that, if M equals N, and N equals P, then M also equals P. The transitive property of inequality states that if M is greater than N and N is greater than P, then M is also greater than P.

The transitive property of inequality also holds true for less than, greater than or equal to, and less than or equal to. The transitive property holds for mathematics, but not always in real settings. For example, just because Team A beat Team B and Team B beat Team C does not mean that Team A will beat Team C. It is important to use the transitive property only in the certain situations, or incorrect conclusions, like team A will beat team C, will be reached. Other important properties of equality include the reflexive and symmetric properties. The reflexive property states that any quantity is equal to itself, so Y equals Y. The symmetric property states that if J equals K, then K equals J. The substitution postulate states that if H equals B, then H may be substituted for B, and B may be substituted for H in any equation. These properties and postulates are used in many Euclidean proofs. The equality properties also hold for vector quantities assuming the vectors are the same size.