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{{Short description|none}}
This article examines the implementation of mathematical concepts in [[set theory]]. The implementation of a number of basic mathematical concepts is carried out in parallel in [[ZFC]] (the dominant set theory) and in [[New Foundations|NFU]], the version of Quine's [[New Foundations]] shown to be consistent by [[R. B. Jensen]] in 1969 (here understood to include at least axioms of [[Axiom of infinity|Infinity]] and [[Axiom of choice|Choice]]).
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The '''downward closure''' of a member <math>x</math> of the field of <math>R</math> is the smallest set <math>D</math> containing <math>x</math>, and containing each <math>zRy</math> for each <math>y \in D</math> (i.e., including the preimage of each of its elements with respect to <math>R</math> as a subset.)
The '''[[relation composition|relative product]]''' <math>R
Notice that with our formal definition of a binary relation, the range and codomain of a relation are not distinguished. This could be done by representing a relation <math>R</math> with codomain <math>B</math> as <math>\left(R, B\right)</math>, but our development will not require this.
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