"In mathematics, the := =: and = symbols are used for definition."

[StoneCypher]

In mathematics, the := =: and = symbols are used for definition.

Are all three equivalent? If not, would you kindly explain the differences? If so, would you mention please that they are exchangeable?

Thank you _very much_ for writing this. I have wanted a document like this forever.

[OscarYuen]

:= means defining a new function/variable
example,
Ax+B := f(x) , where A and B are constant
we could say that we define f(x) as Ax+B

=: I've seen it before. I guess it is the same as above but reverse direction

= means equivalence

[mattdesl]

It depends a bit on the literature. Some will do it like this.¹

A := B

A is defined as B

Some literature uses = to also mean :=.

[debmanna]

• " = " (the equals sign) means “is the same as” and was first introduced in the 1557 book The Whetstone of Witte by Robert Recorde (c. 1510-1558).
• " := " (the equal by definition sign) means “is equal by definition to”. This is a common alternate form of the symbol “=Def”, which appears in the 1894 book Logica Matematica by the logician Cesare Burali-Forti (1861–1931). Other common alternate forms of the symbol “=Def” include“def = ” and “≡”, the latter being especially common in applied mathematics.

Source: Some Common Mathematical Symbols and Abbreviations (with History) Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 21, 2007)

Summary:

• Use " = " when you are simplifying some expression or equating two functions
  example :

1. (x+2)(x+1) = x^2+3x+2
2. 2^2 = 4

• Use " := " / “≡” to imply equivalence or when you are assigning an expression to a new variable say
  example:

1. f(x) := x^2+3x+2