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Let A € R2. Let B € R be defined by B = {x € R there is y € R such that (x,Y) € A}_B is called the projection of A onto the x-axis_ (a...

Question

Let A € R2. Let B € R be defined by B = {x € R there is y € R such that (x,Y) € A}_B is called the projection of A onto the x-axis_ (a) If A is a closed set in R2, is B necessarily a closed set in R? (b) If A is an open set in R2,is B necessarily an open set in R?

Let A € R2. Let B € R be defined by B = {x € R there is y € R such that (x,Y) € A}_ B is called the projection of A onto the x-axis_ (a) If A is a closed set in R2, is B necessarily a closed set in R? (b) If A is an open set in R2,is B necessarily an open set in R?



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Determine whether the given set $S$ of vectors is closed under addition and closed under scalar multiplication. In each case, take the set of scalars to be the set of all real numbers. The set $S:=\left\{(x, y) \in \mathbb{R}^{2}: y=x^{2}\right\}.$


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