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:
“The consequences of our actions grab us by the scruff of our necks, quite indifferent to our claim that we have “gotten better” in the meantime.”
—Friedrich Nietzsche (1844–1900)
“The horror of Gandhi’s murder lies not in the political motives behind it or in its consequences for Indian policy or for the future of non-violence; the horror lies simply in the fact that any man could look into the face of this extraordinary person and deliberately pull a trigger.”
—Mary McCarthy (1912–1989)
“War is thus divine in itself, since it is a law of the world. War is divine through its consequences of a supernatural nature which are as much general as particular.... War is divine in the mysterious glory that surrounds it and in the no less inexplicable attraction that draws us to it.... War is divine by the manner in which it breaks out.”
—Joseph De Maistre (1753–1821)