By Nicolas Bourbaki (auth.)

This is a softcover reprint of the English translation of 1968 of N. Bourbaki's, Théorie des Ensembles (1970).

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Intuitively, l: represents the unique object which has the property defined by R. • For example, in a theory where "y is a real number ~ 0" is a theorem, the relation "x is a real number ~ 0 and y = x·" is functional in x. The corresponding functional symbol is taken to be either or yl/2•• Vy C47. Let x be a letter which is not a constant of 'G, and let R Ix! and S I x ! be two relations in 'G. If R Ix! is functional in x in 'G, then the relation SI"'x(R)! is equivalent to (3x)(Rlx! ). ) is equivalent to (Rlxl and SI"'x(R) i); since SI'tx(R)!

Y I = J'! ) ~ (J'\yl = J'\zj), (J'ly! ), is true. Now, I is true by Theorem 1; hence J'! = J'l z I is true. 3 FUNCTIONAL RELATIONS From all this it follows that (y = z) ==> (V I y! = V! z I) is a theorem in '(00' say A. But (Tly)(Ulz)A is precisely A relation of the form T = U, where T and U are terms in '(0, is called an equation; a solution (in '(0) of the relation T = U, considered as an equation in a letter x, is therefore (§2, no. 2) a term V in '(0 such that T I V! = U I V! is a theorem in '0.

Now, since R is singlevalued in x, (R and ("tJ[(R)lx)R) ==> (x = "tJ[(R» is a theorem in 'CO by C30 (§4, no. 3). Therefore x = "tJ[(R) is true. ~ Conversely, suppose that R ==> (x = T) is a theorem in 'CO. Let y, z be distinct letters which are distinct from x and which appear neither in R nor in the explicit axioms of 'CO. Since x is not a constant of 'CO and does not appear in T, the relations (Ylx)R ==> (y = T), (zlx)R==> (z = T) are theorems in to. Adjoin the hypotheses (ylx)R and (zlx)R. y = T and z = T are true, hence y = z is true.

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