The Numerous Interpretations and Implications of AB in Mathematics and Beyond

Imagine an equation as a balance scale:

If a is 5 and b is 5, a b, and 5 5. Equal means balanced.

5 5 10

Here, the expression on the right and left of the equation can be visualized as weights on a scale. When weighed, the scale is balanced, indicating a balanced equation.

Formal Definitions of AB

Mathematically, the equality sign represents more than a simple balance; it often carries rich underlying implications. Here, we present three suggestions:

1. Identity Relation for Sets

A B Leftrightarrow A subset B land B subset A

This definition describes the identity relation for sets. It signifies that for two sets to be equal, they must be subsets of each other.

2. Extensionality Principle for Identity

A B Leftrightarrow forall a (a in A leftrightarrow a in B)

This interpretation states that two sets are equal if and only if every element of one set is also an element of the other set and vice versa.

3. Principle of the Identity of Indiscernibles

A B Leftrightarrow forall P (A in P leftrightarrow B in P)

Here, equality is said to hold if and only if, for every property or predicate P, A has that property if and only if B has that property.

Equality in Context

Equality within a specific context can lead to various interpretations. For instance, in financial transactions, 1 dollar is equal to 4 quarters. However, in the context of a coin-operated washer in an apartment building, this statement is not true, as the machine only accepts quarters.

Consider counting babies in a maternity ward. Upon saying, “Baby-number-one,” it is implied that there is a one-to-one correspondence between the label and the baby itself. This context-dependent interpretation of equality highlights the importance of the language and context in which an equality is stated.

For many programmers, A B implies an assignment operation, meaning that the value of variable B is assigned to variable A. Thus, A B and B A represent two distinct statements in this context.

Mathematical Equality

In mathematical terms, 4 4 is straightforward. However, when statements like 1/2 2/4 are made, they already involve assumptions. Strictly speaking, these are two different strings of symbols, but they represent the same number because 1/2 is equivalent to 2/4.

Note: Equality is not always easy. If you attempt to define it rigorously, you'll find it is an equivalence relation with a partial order and is actually the smallest possible equivalence relation on any set of objects. Assigning equality to different objects can be a dangerous path.

Equivalence Classes

Mathematical equality can be further abstracted into equivalence classes. For example, two numbers a and b are equal if a mod 7 b mod 7. This is a more specific equality relation, saying that they leave the same remainder after division by 7.

Isomorphism and Identity

Two sets {x in | x2 2} and {-101} are equal because they represent the same objects in different presentations. However, sets {a, b, c} and {1, 2, 3} are isomorphic, meaning they have the same structure but are not identical. There are no better mappings between these sets than another.

Equality, therefore, is a nuanced concept deeply rooted in the context and the underlying assumptions about the objects in question.

Conversational and Abstract Use of Equality

At its core, the concept of equality is not just about numerical or set-theoretic equivalence but often represents a deeper metaphorical or contextual similarity.

For example, if you're in a desert and need water, a canteen of water and a questionable puddle of slush can be considered equal in terms of quenching thirst. Here, equality is not about physical identity but about the contexts that deem the objects as functionally equivalent.

Note: Barry Mazur’s treatment of mathematical equality is a great resource for those who wish to delve deeper into this topic, as it discusses these nuanced aspects of equality in depth.