Suppose f X rightarrow Y and A B Y Prove that A B implies f1

Suppose f: X rightarrow Y and A, B Y. Prove that A B implies f^-1(A) f^-1(B). f^-1(A B) = f^-1(A) f^-1(B). f^-1(A B) = f^-1(A) f^-1(B).

Solution

(A)Suppose f:X->Y and A,B Y

then,to prove

AB implies f-1(A)f-1(B) can easily be concluded..

further,

Let f : X Y be an arbitrary function and A,B Y.

(B) Proof.   

Let x f ¹ (A B).

Then either f(x) A or f(x) B;

in the first case x f ¹ (A), while in the second case x f ¹ (B).

Either way x f ¹ (A) f ¹ (B),

whence,

f ¹ (A B) f ¹ (A)f ¹ (B).

Now, let y f ¹ (A) f ¹ (B).

Then either f(y) A or f(y) B.

Either way, f(y) A B,

so we deduce y f¹ (A B) and f ¹ (A B) = f ¹ (A) f ¹(B)

(C). Proof.

Let x f ¹ (A B),

so that, f(x) A B.

Then f(x) A and f(x) B;

that is, x f ¹ (A) and x f ¹ (B).

From this we deduce f ¹ (A B) f ¹ (A) f ¹ (B).

Now assume y f ¹ (A) f ¹ (B).

Since y f ¹ (A) we have f(y) A. Since y f ¹ (B) we have f(y) B.

Hence we have f(y) A B and y f ¹ (A B).

We conclude that f ¹ (A B) = f ¹ (A) f ¹ (B).