authored by Premmi and Beguène
Introduction
In mathematics, the relation “equals” is a foundational concept that can be applied to any two mathematical objects to indicate that these objects represent the same mathematical entity. Since the concept of equality pertains to all mathematical objects, it is universal across mathematics and transcends its specific branches, situating itself firmly within the realm of logic. The equality relation is denoted by the symbol “=”.
Examples of Equality Relation
- For numbers:
\hspace*{4cm} 4 + 3 = 7 - For sets:
\hspace*{4cm}\{1, 2, 3\} = \{3, 1, 2\} - For functions:
\hspace*{4cm}f(x) = x^3 = g(x), where g(x) is also defined as x^3 - For matrices:
\hspace*{4cm}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I_2 \, (\text{the } 2 \times 2 \text{ identity matrix}) - For geometric objects:
\hspace*{4cm}\text{The set of all points } (x, y) \in \mathbb{R} \times \mathbb{R} \text{ such that } x² + y² = 1 \, (\text{a circle with radius } 1)
In general, we write a = b to denote that a \text{ and } b represent the same mathematical object. This notation is universal across all areas of mathematics, from basic arithmetic to abstract algebra.
Axioms of Equality
The Axioms of Equality refer to a set of foundational principles that govern the relationship of equality in mathematics. These axioms include two key properties that describe how equality operates among mathematical objects.
In logic, equality is described through the following axioms:
- Law of Identity: This axiom states that any mathematical object is equal to itself, expressed as:
\hspace{6cm}\forall a \,(a = a) - Substitution Axiom: This axiom states that if two mathematical objects are equal, then any property of one must also be a property of the other. This implies that one can be substituted for the other in any mathematical formula without changing the truth of that formula once any free variables in the formula have been instantiated. This axiom is also sometimes referred to as Leibniz’s law.
In mathematical terms this law can be expressed as follows:
For every a \text{ and } b and any formula \phi(x), where x is a free variable, if a = b, then \phi(a) \text{ implies } \phi(b).
Symbolically, this can be represented as:
\hspace{6cm}\forall a, b \,(a = b) \implies \big[\phi(a) \Rightarrow \phi(b)\big]
For example, for all real numbers a \text{ and } b, if a = b, then a \geq 0 \text{ implies } b \geq 0 \, (\text{here } \phi(x) \text{ is } x \geq 0).
It should be noted that these two axioms do not define equality, they only state what properties that objects that are related by equality must satisfy. However, these two axioms are usually sufficient for deducing most properties of equality that mathematicians care about.
We can deduce from these two axioms some more properties of the equality relation. We will discuss these next.
Derivations of Properties of Equality Relation
- Reflexivity of Equality: For any element a in a set S with a relation R induced by equality (xRy \Leftrightarrow x = y), it holds that \forall a \in S\, (aRa).
That is, for any element a in the set S, the relation R asserts that a is related to itself, which is equivalent to saying that a is equal to itself.
Proof. Given some set S with a relation R induced by equality, let us assume that a \in S. Then, by the Law of Identity, a = a and consequently, aRa. - Symmetry of Equality: For any elements a \text{ and } b in a set S with a relation R induced by equality (xRy \Leftrightarrow x = y), it holds that \forall a, b \in S\, (aRb \implies bRa).
That is, for any elements a \text{ and } b in the set S, the relation R asserts that if a is equal to b (i.e., aRb), then it follows that b is equal to a (i.e., bRa).
Proof. Given some set S with a relation R induced by equality, let us assume that there are elements a, b \in S such that aRb. Let us consider the formula \phi(x) : xRa.
By the Substitution Axiom, we have:
\hspace{6cm}(a = b) \implies \big[\phi(a) \Rightarrow \phi(b)\big]
Thus, we obtain:
\hspace{6cm}(a = b) \implies (aRa \Rightarrow bRa)
Since by assumption, a = b and by Reflexivity, aRa, it follows that bRa.
Therefore, we have shown that if aRb, then bRa, demonstrating the symmetry of equality. - Transitivity of Equality: For any elements a, b \text{ and } c in a set S with a relation R induced by equality (xRy \Leftrightarrow x = y), it holds that \forall a, b, c \in S\, \big[(aRb \land bRc) \implies aRc\big].
That is, for any elements a, b \text{ and } c in the set S, the relation R asserts that if a is equal to b (i.e., aRb) and b is equal to c (i.e., bRc), then it follows that a is equal to c (i.e., aRc).
Proof. Given some set S with a relation R induced by equality, let us assume that there are elements a, b, c \in S such that aRb \text{ and } bRc. Then let us consider the formula \phi(x) : xRc.
By the Substitution Axiom,
\hspace{5cm}(b = a) \implies (bRc \Rightarrow aRc).
Since by assumption, a = b, it follows from Symmetry that b = a. Additionally, since by assumption bRc, it follows that aRc.
Therefore, we have shown that if aRb \text{ and } bRc, then aRc, demonstrating the transitivity of equality. - Function Application: For any elements a \text{ and } b from the domain of a function f it holds that if a = b then it implies that f(a) = f(b).
Proof. Given a function f, let us assume that there are elements a \text{ and } b from its domain such that a = b. We will consider the formula \phi(x) : f(a) = f(x).
By the Substitution Axiom, we have:
\hspace{6cm}(a = b) \implies \big[(f(a) = f(a)) \Rightarrow (f(a) = f(b))\big].
Since, by assumption a = b and by the Law of Identity f(a) = f(a), it follows that f(a) = f(b).
Therefore, we have shown that for any elements a \text{ and } b from the domain of a function f, if a = b then it must be the case that f(a) = f(b).
These properties are sometimes included in the axioms of equality, but it is not necessary to include them since they can be derived from the two axioms of equality as shown above.