Question 3. let f: x --> y be a function. (a) recall that for c cy, the inverse image of c is the set f-1(c) = {x e x|f(x) e c}. let a, b cy. prove that f-1(anb) cf-1(a) n f-1(b). (b) recall that for d c x, the image of d under f is the set f(d) = {f(x) e yx e x}. let a, b c x. prove that f(an b) c f(a)n f(b). find an example of a function f and sets a and b such that f(anb) = ø but f(a) n f(b) + ø.

Step-by-step explanation:

a) We want to prove that . Then, we can do that proving that every element of   is an element of too.

Then, suppose that . From the definition of inverse image we know that , which is equivalent to and . But, as we can affirm that and, because   we have .

Therefore, .

b) We want to prove that . Here we will follow the same strategy of the above exercise.

Assume that . Then, there exists such that . But, as we know that and . From this, we deduce and . Therefore, .

c) Consider the constant function for every real number . Take the sets and .

Notice that =Ø, so =Ø. But and , so .

∞ this is your sign