The symbol $=$ means equals.
- $x = y$ means $x$ is the same object as $y$, and is read $x$ equals $y$, or $x$ is equal to $y$.
- $x \ne y$ means $x$ is not the same object as $y$, and is read $x$ is not equal to $y$.
- $a = b$
- $a$ and $b$ are names for the same object.
The word equality is the noun derived from the verb equals.
Note on Terminology
This distinction is often not made.
When such a difference is important the symbol $=$ may be used for equals and $\cong$ for congruent.
- Equals is Equivalence Relation
- Axiom:Leibniz's Law
- Definition:Diagonal Relation
- Definition:Set Equality
- The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.
Placing two hyphens together, one above the other, he wrote:
- "To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: $= \!\!\! = \!\!\! = \!\!\! = \!\!\! =$, bicause noe .2. thynges, can be moare equalle."
The word gemowe comes from the Latin geminus meaning twin.
Both were in due course supplanted by $=$. It is suggested by some sources that this was mainly through the influence of Leibniz.
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) $\S 16: \ 19$
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.1$. Relations on a set
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Equality
- 1972: Patrick Suppes: Axiomatic Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts