Question

Any help is appreciated!! Let f:X-->Y be a function and suppose that A,B subset X. Prove or disprove: If f(A) subset f(B) then A subset B.

Answer 1

Consider \(A \cap B = \varnothing\) and then consider a constant function \(f(A) = f(B) = \{c\}\)

Notice that \(f(A) =\{c\} \subset \{c\} = f(B)\)

Answer 2

Do you see what I mean?

Answer 3

So are you saying consider the intersection of A and B is the empty set. And then consider f(A) is equal to f(B) and say they are equal to c.
Then are you saying f(A) = c and c = f(B)?

Answer 4

Just think about any constant function, f(x) = c for all \(x\). Notice that the image of any two disjoint sets under this map will be \(\{c\}\). Do you see that?

Answer 5

Let \(f :\{1, 2\} \to \mathbb{N}\) and let \(f(\{1\}) = f(\{2\}) = \{c\}\) certainly \(\{1\} \not\subset \{2\}\) but \(\{c\} \subset \{c\}\)

Answer 6

thank you!