Pathwise connected
Give an example: Let $A$ and $B$ be pathwise-connected subsets of $\mathbb{R}^2$, whose intersection $A∩B$ is nonempty. The union $A∪B$ is pathwise connected.
*Use the fact that a convex subset is pathwise connected for one of the subsets.
Example:
$A=\{(x,y)\in \mathbb{R}^2 | \hspace{1mm} |x|+|y| \leq 10\}$
$B=\{(x,y)\in \mathbb{R}^2 | \hspace{1mm} x^2+y^2=25\}$
and let $C=A \cup B$. Define the function $f:C\rightarrow \mathbb{R}$ by
$f(x,y)=\frac{5x+2y}{{\sqrt{x^2+y^2}}+1}$
Something similar to this example. But I need different subsets and function from this.
Imanonymous
Answer
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Mathe
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Sorry can I request for a more complicated subsets? Just like the example I have given. I wish to have something like x^2+y^2=25 (e.g. a shape)
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You can take A= {(x,y)| y>=x^2}, this is everything above the parabola y = x^2 and it is path connected. Take B = {(x,y)| y<=1-x^2}, this is everything below the parabola y = 1- x^2 . The intersection is not empty and their union is path connected.
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Or can you help me to verify if I can define the two subsets as: A=[-3,3]x[-4,4] which is a generalised triangle and B={(x,y) | x^2+y^2=4}.
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May I have some steps to show that y>=x^2 by using the definition of pathwise connected?
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I also need a function with AUB as its domain, as shown in the example :)
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The function sin(x+y) is defined on AUB.
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The definition of path connected means that every pair of points in the set can be joined by a path. In this case, a line between two points will join them.
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Can I have a function that is more complicated?
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Sin((5x^2-2y^3)/(1+^x4+y^6))
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Thank you!
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Is each of these subsets is pathwise-connected? A=[-3,3]x[-4,4] which is a generalised triangle and B={(x,y) | x^2+y^2=4}
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A=[-3,3]x[-4,4] is just a square, but yeah, they are path-connected.
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Thank you for your help!
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Are you sure you wrote the question correctly?
Which part is wrong?
Well, you could take A = B = R^2. All sets are convex and path-connected, A intersection B = R^2 is non empty and A union B = R^2 which is again convex and path-connected. One could also find easy examples with A different to B.
I have edited the question to make it clearer.
Please fix: " Let A and B of be pathwise-connected of \mathbb{R}^2". That statement is not correct.
Sorry for the mistake.
The bounty is low.
You just need an example or you also need a proof that it satisfied the conditions?
A simple proof to show that each two subsets are pathwise connected or just hint would be fine.
I mean each subsets
I'm confused now. Do you need to prove that, "If A and B are path connected with non-empty intersection, THEN their union is also path connected"? Or do you need an example of this behavior? If it's just the example, I can answer that very quickly.
Just the example with explanation.