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)
Famous quotes containing the word consequences:
“If you are prepared to accept the consequences of your dreams ... then you must still regard America today with the same naive enthusiasm as the generations that discovered the New World.”
—Jean Baudrillard (b. 1929)
“There are more consequences to a shipwreck than the underwriters notice.”
—Henry David Thoreau (1817–1862)
“We are still barely conscious of how harmful it is to treat children in a degrading manner. Treating them with respect and recognizing the consequences of their being humiliated are by no means intellectual matters; otherwise, their importance would long since have been generally recognized.”
—Alice Miller (20th century)