1. ## Solution Set Question.

Hi,

I'm new to the forum but hope to contribute when and where I can.

I'm doing some study at the moment and I was stuck on the following problem.

Help appreciated.

Let the solution set of f(x)=0 be A and the solution set of g(x)=0 be B (here f and g are real functions taking real values).

What is the solution sets of f(x)g(x)=0 and of f(x)2+g(x)2=0.

I'm not sure how this answer is supposed to look.

My initial thoughts were something along the lines of

f(x).g(x) = A.B = 0 [A product B]

and

f(x)^2 + g(x)^2 = A^2 + B^2 = 0

But I really don't know if this is going in the right direction at all?

If you have a minute I would be grateful for your thoughts.

-Infinite.

2. f(x).g(x) = A.B = 0 [A product B]
This does not make type sense, or at least more explanation is needed. What do you mean by the product of two sets: the Cartesian product or maybe $\{xy\mid x\in A, y\in B\}$? What does 0 mean: is it a number? Then how can a product of two sets equal a number? Similarly, for what x do you consider f(x)g(x), or is it some product of functions as a whole?

Try to find a relationship between the propositions f(x) = 0 and f(x)g(x) = 0. Which implies which? What does it say about the relationship between A and the solution set of f(x)g(x) = 0?

3. It odd that just this week I have seen the old notation for union and intersection come up on this board. R.L. Moore and many of his students used $AB\text{ or }A\cdot B$ for $A\cap B$ and $A+B$ for $A\cup B$.
Originally Posted by infinitepersepctives
Let the solution set of f(x)=0 be A and the solution set of g(x)=0 be B (here f and g are real functions taking real values).
What is the solution sets of f(x)g(x)=0 and of f(x)2+g(x)2=0.
For the solution set of $f(x)g(x)$ it would be $A\cup B$.

For the solution set of $f^2(x)+g^2(x)$ it would be $A\cap B$.

4. Thanks for that. Is there a possibility of a short explanation on this?

For the solution set of $f(x)g(x)$ it would be $A\cup B$.

For the solution set of $f^2(x)+g^2(x)$ it would be $A\cap B$.
Are these well-known rules rather than a problem exercise?

5. It is common sense.
$A\cdot B=0$ if and only if $A=0\text{ or }B=0$. OR is union.

$A^2+ B^2=0$ if and only if $A=0\text{ and }B=0$. AND is intersection.

6. Thank you! That makes sense alright! :-) Great!