🔗 Definition 5.5. 🔗 🔗Let .P⊆A. The image of P under ,f, denoted ,f(P), is the set defined by f(P)=b∈B|b=f(p)forsomep∈P. 🔗The range of f (also called the image of f ) is the set .f(a).
🔗 Definition 5.6. 🔗 🔗Let .Q⊆B. The inverse image of Q under ,f, denoted ,f−1(Q), is the set defined by f−1(Q)=a∈A|f(a)∈Q.
🔗 Theorem 5.7. 🔗 🔗Let A and B be sets, let C,D⊆A and S,⊓⊆B be subsets, and let f:A→B be a function. Let I and K be non-empty sets, let Uii∈I be a family of subsets of A indexed by ,I, and let Vkk∈K be a family of subsets of B indexed by .K. 🔗 f(∅)=∅ and .f−1(∅)=∅. 🔗.f−1(B)=A. 🔗 f(C)⊆S if and only if .C⊆f−1(S). 🔗If ,C⊆D, then .f(C)⊆f(D). 🔗If ,S⊆T, then .f−1(S)⊆f−1(T). 🔗.f(⋃i∈IUi)=⋃i∈If(Ui). 🔗.f(⋂i∈IUi)⊆⋂i∈If(Ui). 🔗.f−1(⋃k∈KVk)=⋃k∈Kf−1(Vk). 🔗.f−1(⋂k∈KVk)=⋂k∈Kf−1(Vk).
Let A and B be sets, let P,Q⊆A be subsets and let f:A→B be a function. Prove that .f(P)−f(Q)⊆f(P−Q). Is it necessarily the case that ?f(P−Q)⊆f(P)−f(Q)? Give a proof or a counterexample.
