3 8 points define a binary relation e on r2 by lefta_1 a_2right eleftb

Question

3. (8 points) Define a binary relation E on R² by \left(a_{1}, a_{2}\right) E\left(b_{1}, b_{2}\right) \Longleftrightarrow a_{1}^{2}+a_{2}^{2}=b_{1}^{2}+b_{2}^{2} You may take for granted the fact that E is an equivalence relation on R². (a) (4 points) What is the equivalence class [(1,0)]E? What kind of geometric object is[(1,0)]E in the real plane R^2? \text { (b) (6 points) Define a function } f: \mathbf{R}^{2} \times \mathbf{R}^{2} \rightarrow \mathbf{R}^{2} \text { by } f\left(\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right)\right)=\left(\sqrt{a_{1}^{2}+a_{2}^{2}}+\sqrt{b_{1}^{2}+b_{2}^{2}}, 0\right) \text { for }\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right) \in \mathbf{R}^{2} . \text { Prove that } f \text { respects the equivlenence relation } E \text { . }