sign in two different ways, as noted by the quote from PM:✸13.01. x = y .=: (φ): φ ! x . ⊃ . φ ! y Dfmeans: "This definition states that x and y are to be called identical when every predicative function satisfied by x is also satisfied by y ... Note that the second sign of equality in the above definition is combined with "Df", and thus is not really the same symbol as the sign of equality which is defined." The not-equals sign "≠" makes its appearance as a definition at ✸13.02.✸14: Descriptions: "A description is a phrase of the form "the term y which satisfies φŷ, where φŷ is some function satisfied by one and only one argument." From this PM employs two new symbols, a forward "E" and an inverted iota "℩". Here is an example:✸14.02. E ! ( ℩y) (φy) .=: ( Ǝb):φy . ≡y . y = b Df. This has the meaning: "The y satisfying φŷ exists," which holds when, and only when φŷ is satisfied by one value of y and by no other value." (PM 1967:173–174) Introduction to the notation of the theory of classes and relations The text leaps from section ✸14 directly to the foundational sections ✸20 GENERAL THEORY OF CLASSES and ✸21 GENERAL THEORY OF RELATIONS. "Relations" are what is known in contemporary set theory as sets of ordered pairs. Sections ✸20 and ✸22 introduce many of the symbols still in contemporary usage. These include the symbols "ε", "⊂", "∩", "∪", "–", "Λ", and "V": "ε" signifies "is an element of" (PM 1962:188); "⊂" (✸22.01) signifies "is contained in", "is a subset of"; "∩" (✸22.02) signifies the intersection (logical product) of classes (sets); "∪" (✸22.03) signifies the union (logical sum) of classes (sets); "–" (✸22.03) signifies negation of a class (set); "Λ" signifies the null class; and "V" signifies the universal class or universe of discourse. Small Greek letters (other than "ε", "ι", "π", "φ", "ψ", "χ", and "θ") represent classes (e.g., "α", "β", "γ", "δ", etc.) (PM 1962:188): x ε α "The use of single letter in place of symbols such as ẑ(φz) or ẑ(φ ! z) is practically almost indispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus ' x ε α' will mean ' x is a member of the class α'". (PM 1962:188) α ∪ –α = V The union of a