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:
“[As teenager], the trauma of near-misses and almost- consequences usually brings us to our senses. We finally come down someplace between our parents’ safety advice, which underestimates our ability, and our own unreasonable disregard for safety, which is our childlike wish for invulnerability. Our definition of acceptable risk becomes a product of our own experience.”
—Roger Gould (20th century)
“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)
“Resistance is feasible even for those who are not heroes by nature, and it is an obligation, I believe, for those who fear the consequences and detest the reality of the attempt to impose American hegemony.”
—Noam Chomsky (b. 1928)