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
Two objects being equal is not necessarily the same as two objects being congruent.
This distinction is often not made.
When such a difference is important the symbol $=$ may be used for equals and $\cong$ for congruent.
- Equality is Equivalence Relation
- Axiom:Leibniz's Law
- Definition:Diagonal Relation
- Definition:Set Equality
The equals sign was introduced by Robert Recorde in his $1557$ work The Whetstone of Witte.
Placing two long 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.
François Viète used the symbol $\sim$, while René Descartes used $\propto$.
Both were in due course supplanted by $=$, a shortened and hence more efficient version of Recorde's invention.
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 $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 6$ Significance of the results