Quotient spaces and equivalence relations
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Exercise:
Let v_ v_ in V we define v_ sim v_ iff v_-v_ in U. Show that sim defines an equivalence relation on V.
Solution:
Proof of refelxivity. vsim v because v-vin U. Proof of symmetry. If v_sim v_ Longrightarrow v_sim v_ because v_-v_-v_-v_in U. Proof of transitivity. If v_sim v_ v_sim v_ Longrightarrow v_sim v_ Indeed because v_-v_v_-v_+v_-v_in U. We denote the equivalence class of vin V by v. vv+Uv+u|uin U.
Let v_ v_ in V we define v_ sim v_ iff v_-v_ in U. Show that sim defines an equivalence relation on V.
Solution:
Proof of refelxivity. vsim v because v-vin U. Proof of symmetry. If v_sim v_ Longrightarrow v_sim v_ because v_-v_-v_-v_in U. Proof of transitivity. If v_sim v_ v_sim v_ Longrightarrow v_sim v_ Indeed because v_-v_v_-v_+v_-v_in U. We denote the equivalence class of vin V by v. vv+Uv+u|uin U.