Equivalence is a relation that holds between constructs that are the “same” in some context. Two constructs are equivalent if they are similar for one or more reasons. The basic symbols that we use for equivalence are _ and P, and the related l.
Equivalence is a context-sensitive relation. Consider the two strings “Y = 2 _ X” and “Z = X _ 2”. Textually, they are not equivalent because they contain different characters. If interpreted in an algebraic setting, they are potentially equivalent equations, but it depends upon the specific algebras in question. If they both are interpreted in the same algebra, if that algebra is commutative, and alpha-renaming is a part of the definition of equivalence, then they are equivalent. If any of these conditions fails, or perhaps some other unusual conditions exist (i.e., one algebra is a modulus group), then the equations are not equivalent. Equivalence is not just something that is applicable to formal mathematical statements. Consider a pinball
machine and a Sony PlayStation. How are these two things equivalent? Obviously, both are games that people play for enjoyment, so this classification criterion is one potential equivalence class. Another equivalence is that both devices have buttons, thus some property-based criterion imply equivalence. Another more subtle equivalence class is the fact that both devices encode the group (Z, +) Because both keep score in some fashion.
