Image (mathematics) - Consequences

Consequences

Given a function f : XY, for all subsets A, A1, and A2 of X and all subsets B, B1, and B2 of Y we have:

  • f(A1A2) = f(A1) ∪ f(A2)
  • f(A1A2) ⊆ f(A1) ∩ f(A2)
  • f −1(B1B2) = f −1(B1) ∪ f −1(B2)
  • f −1(B1B2) = f −1(B1) ∩ f −1(B2)
  • f(A) ⊆ BAf −1(B)
  • f(f −1(B)) ⊆ B
  • f −1(f(A)) ⊇ A
  • A1A2f(A1) ⊆ f(A2)
  • B1B2f −1(B1) ⊆ f −1(B2)
  • f −1(BC) = (f −1(B))C
  • (f |A)−1(B) = Af −1(B).

The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:

(Here, S can be infinite, even uncountably infinite.)

With respect to the algebra of subsets, by the above we see that the inverse image function is a lattice homomorphism while the image function is only a semilattice homomorphism (it does not always preserve intersections).

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