Question

Let f : S → T be a function. Suppose A and B be subsets of S,

and C and D be subsets of T .
c) Prove that A ⊆ f −1[f [A]].
Provide a counterexample to show that f −1[f [A]] is not
a subset of

Answer 1

To prove that A ⊆ f-1[f[A]], we need to show that every element in A is also in f-1[f[A]]. A counterexample can be provided to show that f-1[f[A]] is not necessarily a subset of A.

Step-by-step explanation:

To prove that A ⊆ f-1[f[A]], we need to show that every element in A is also in f-1[f[A]]. Suppose x is an element in A. Then f(x) is an element in f[A]. Since f(x) is in f[A], it follows that x must be in f-1[f[A]]. Therefore, A ⊆ f-1[f[A]].

To provide a counterexample to show that f-1[f[A]] is not necessarily a subset of A, consider the function f: {1, 2} → {1} defined by f(1) = f(2) = 1. Let A = {1} and B = {}. In this case, f-1[f[A]] = {1, 2} but A = {1}. Therefore, f-1[f[A]] is not a subset of A in this example.