Consequences
Given a function f : X → Y, for all subsets A, A1, and A2 of X and all subsets B, B1, and B2 of Y we have:
- f(A1 ∪ A2) = f(A1) ∪ f(A2)
- f(A1 ∩ A2) ⊆ f(A1) ∩ f(A2)
- f −1(B1 ∪ B2) = f −1(B1) ∪ f −1(B2)
- f −1(B1 ∩ B2) = f −1(B1) ∩ f −1(B2)
- f(A) ⊆ B ⇔ A ⊆ f −1(B)
- f(f −1(B)) ⊆ B
- f −1(f(A)) ⊇ A
- A1 ⊆ A2 ⇒ f(A1) ⊆ f(A2)
- B1 ⊆ B2 ⇒ f −1(B1) ⊆ f −1(B2)
- f −1(BC) = (f −1(B))C
- (f |A)−1(B) = A ∩ f −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).
Read more about this topic: Image (mathematics)
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