Suppose that set ( A ) is equivalent to set ( B= left {(x, y) in mathbb{R}^{2}: x^{2}+y^{2}=1 right } ). Let ( C= left {(x, y) in mathbb{R}^{2}: left( frac{x}{2} right)^{2}+ left( frac{y}{3} right)^{2}=1 right } ) (1) [2 points] Prove that if mappings ( f: A longrightarrow B ) and ( g: B longrightarrow C ) are injective, then the

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Suppose that set  ( A  ) is equivalent to set  ( B= left {(x, y)  in  mathbb{R}^{2}: x^{2}+y^{2}=1 right }  ). Let  ( C= left {(x, y)  in  mathbb{R}^{2}: left( frac{x}{2} right)^{2}+ left( frac{y}{3} right)^{2}=1 right }  ) (1) [2 points] Prove that if mappings  ( f: A  longrightarrow B  ) and  ( g: B  longrightarrow C  ) are injective, then the composition  ( g  circ f: A  longrightarrow C  ) must be injective. (2) [2 points] Prove that if mappings  ( f: A  longrightarrow B  ) and  ( g: B  longrightarrow C  ) are surjective, then the composition  ( g  circ f: A  longrightarrow C  ) must be surjective. (3) [6 points] Prove or disprove that sets  ( A  ) and  ( C  ) are equivalent. If you think that the statement is true, then make use of part (1) and part (2) above. Construct mapping  ( g: B  longrightarrow C  ).

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(Solved): Suppose that set \( A \) is equivalent to set \( B=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1 ...

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