1. ## F(a-b)=f(a)-f(b)

Let X and Y be sets, and let A and B be any subsets of X.Determine if for all functions from X to Y, F(A-B) = F(A) - F(B) Justify your answer

intuition tells me no because the F(A-B) will have a different x values going to a different y values in Y than F(A) - F(B)

also,

the left side will have x values from X such that they are in $A \cap B^c$

whereas the right side would have x values from X such that they are in $A \cup B$

and clearly $A \cap B^c \neq A \cup B$

2. ## Re: F(a-b)=f(a)-f(b)

maybe I should do a counterexample along with it

let A = {1,2,3,4,5} and B = {1,3,5,6,7}

and have the function y = x^2

then
$F(A-B) = {4,16}$

$F(A) - F(B) = {0,-8,-24,-35,-48......}$

3. ## Re: F(a-b)=f(a)-f(b)

Originally Posted by Jonroberts74
maybe I should do a counterexample along with it

let A = {1,2,3,4,5} and B = {1,3,5,6,7}

and have the function y = x^2

then
$F(A-B) = {4,16}$
To get "{" and "}" in latex, you need to use "\{" and "\}"
$F(A- B)= F(\{2, 4\})= \{4, 16\}$
Yes, that is correct.

$F(A) - F(B) = {0,-8,-24,-35,-48......}$
But now you have suddenly changed your definition of "-". For sets "X- Y" is defined as "all those members of X that are not in Y". It does NOT mean "subtract members of Y from members of X"!
$F(A)= \{1, 4, 9, 16, 25\}$. $F(B)= \{1, 9, 25, 36, 49\}$

So $F(A)- F(B)= \{4, 16\}$ again. This is not a "counter-example".

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# f(a)-f(b)

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