Prove
(A∪B)UC = AU(BUC)

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{If} \mathrm{U} \mathrm{is} \mathrm{universal} \mathrm{set} \mathrm{then}, \\ \mathrm{P}\cup \mathrm{Q} \mathrm{implies} \mathrm{that} \left\{\mathrm{x}\in \mathrm{U};\mathrm{x}\in \mathrm{P} \mathrm{or} \mathrm{x}\in \mathrm{Q}\right\} \\ \mathrm{or} \\ \mathrm{P}\cup \mathrm{Q} = \left\{\mathrm{x}\in \mathrm{U};\mathrm{x}\in \mathrm{P} \mathrm{or} \mathrm{x}\in \mathrm{Q}\right\} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=\left(\mathrm{A}\cup \mathrm{B}\right)\cup \mathrm{C} \\ \Rightarrow \mathrm{L}.\mathrm{H}.\mathrm{S}.= \left\{\mathrm{x}\in \mathrm{U};\mathrm{x}\in \left(\mathrm{A}\cup \mathrm{B}\right) \mathrm{or} \mathrm{x}\in \mathrm{C}\right\} \\ \mathbf{\Rightarrow }\mathbf{L}\mathbf{.}\mathbf{H}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{=}\mathbf{ }\left\{\mathbf{x}\mathbf{\in }\mathbf{U}\mathbf{;}\mathbf{x}\mathbf{\in }\mathbf{A}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{\in }\mathbf{B}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{\in }\mathbf{C}\right\} \\ \mathrm{R}.\mathrm{H}.\mathrm{S}.=\mathrm{A}\cup \left(\mathrm{B}\cup \mathrm{C}\right) \\ \Rightarrow \mathrm{R}.\mathrm{H}.\mathrm{S}.= \left\{\mathrm{x}\in \mathrm{U};\mathrm{x}\in \mathrm{A} \mathrm{or} \mathrm{x}\in \left(\mathrm{B}\cup \mathrm{C}\right)\right\} \\ \mathbf{\Rightarrow }\mathbf{R}\mathbf{.}\mathbf{H}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{=}\mathbf{ }\left\{\mathbf{x}\mathbf{\in }\mathbf{U}\mathbf{;}\mathbf{x}\mathbf{\in }\mathbf{A}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{\in }\mathbf{B}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{\in }\mathbf{C}\right\} \\ \mathbf{\Rightarrow }\mathbf{L}\mathbf{.}\mathbf{H}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{=}\mathbf{R}\mathbf{.}\mathbf{H}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{ }\left(\mathbf{Hence}\mathbf{ }\mathbf{proved}\right) \end{gathered}$$

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