Axiom Schema of Replacement

Set Theory · Axiom Academy

LESSON Axiom Schema of Replacement The image of a set under a function is a set If we have a set A and a function F that assigns to each element x in A a unique value F(x), then the collection of all these F(x) values forms a set. Think of it as: "If you can map every element of a set to something, the collection of those 'somethings' is also a set." The Axiom of Replacement is actually an axiom schema because we need one axiom for each possible formula F(x,y) that defines a functional relation. A functional relation means: for each x, there is at most one y such that F(x,y) holds. When we write F(x,y), we're saying "y is the result of applying F to x." For every set A and every functional formula F(x,y), there exists a set B containing exactly the images of elements from A under F. In symbols: B = y : ∃x ∈ A, F(x,y) . This is the image of A under the functional relation F.

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