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Question Number 93567 by oustmuchiya@gmail.com last updated on 13/May/20

$$GiventhatA=\{0,1,3,5\}B=\{1,2,4,7\}andC=\{1,2,3,5,8\}provethat$$$$\left(\mathrm{i}\right)(\mathrm{A}\cap \mathrm{B})\cap \mathrm{C}=\mathrm{A}\cap (\mathrm{B}\cap \mathrm{C})$$$$\left(\mathrm{ii}\right)(\mathrm{A}\cup \mathrm{B})\cup \mathrm{C}=\mathrm{A}\cup (\mathrm{B}\cup \mathrm{C})$$$$\left(\mathrm{iii}\right)(\mathrm{A}\cup \mathrm{B})\cap \mathrm{C}=(\mathrm{A}\cup \mathrm{C})\cup (\mathrm{B}\cap \mathrm{C})$$$$\left(\mathrm{iv}\right)(\mathrm{A}\cap \mathrm{C})\cup \mathrm{B}=(\mathrm{A}\cup \mathrm{B})\cap (\mathrm{C}\cup \mathrm{B})$$

Commented by Ritu Jain last updated on 13/May/20

$$\left(\mathrm{i}\right)(\mathrm{A}\cap \mathrm{B})\cap \mathrm{C}=\mathrm{A}\cap (\mathrm{B}\cap \mathrm{C})$$$$(\mathrm{A}\cap \mathrm{B})=\{0,1,3,5\}\cap \{1,2,3,7\}=\left\{1\right\}$$$$\mathrm{LHS}=(\mathrm{A}\cap \mathrm{B})\cap \mathrm{C}=\left(\left\{1\right\}\right)\cap \{1,2,3,5,8\}=\left\{1\right\}$$$$$$$$\mathrm{B}\cap \mathrm{C}=\{1,2,4,7\}\cap \{1,2,3,5,8\}=\left\{1\right\}$$$$\mathrm{RHS}=\mathrm{A}\cap (\mathrm{B}\cap \mathrm{C})=\{0,1,3,5\}\cap (\left\{1\right\}=\left\{1\right\}$$$$\mathrm{LHS}=\mathrm{RHS}$$

Answered by otchereabdullai@gmail.com last updated on 14/May/20

$$\mathrm{ii}.\left(\mathrm{AUB}\right)=\{0,1,2,3,4,5,7\}$$$$\mathrm{C}=\{1,2,3,5,8\}$$$$\therefore \left\{\mathrm{AUB}\right\}\mathrm{UC}=\{0,1,2,3,4,5,7,8\}$$$$\mathrm{Also}$$$$\mathrm{A}=\{0,1,3,5\}$$$$\left\{\mathrm{BUC}\right\}=\{1,2,3,4,5,7,8\}$$$$\therefore \mathrm{AU}\left(\mathrm{BUC}\right)=\{0,1,2,3,4,5,7,8\}$$$$\mathrm{Now}\mathrm{since}$$$$\left(\mathrm{AUB}\right)\mathrm{UC}=\mathrm{AU}\left(\mathrm{BUC}\right)=\{0,1,2,3,4,5,7,8\}$$$$\mathrm{we}\mathrm{can}\mathrm{conclude}\mathrm{that}$$$$\left(\mathrm{AUB}\right)\mathrm{UC}=\mathrm{AU}\left(\mathrm{BUC}\right).$$$$$$$$$$$$\mathrm{iii}.\left(\mathrm{AUB}\right)\cap \mathrm{C}=\left(\mathrm{AUC}\right)\mathrm{U}(\mathrm{B}\cap \mathrm{C})$$$$\left(\mathrm{AUB}\right)=\{0,1,2,3,4,5,7\}$$$$\mathrm{C}=\{1,2,3,5,8\}$$$$\therefore \left(\mathrm{AUB}\right)\cap \mathrm{C}=\{0,1,2,3,4,5,7\}\cap \{1,2,3,5,8\}$$$$=\{1,2,3,5\}$$$$\mathrm{also}$$$$\left(\mathrm{AUC}\right)=\{0,1,2,3,5,8\}$$$$(\mathrm{B}\cap \mathrm{C})=\{1,2\}$$$$\therefore \left(\mathrm{AUC}\right)\mathrm{U}(\mathrm{B}\cap \mathrm{C})=\{0,1,2,3,5,8\}$$$$\Rightarrow \left(\mathrm{AUB}\right)\cap \mathrm{C}\ne \left(\mathrm{AUC}\right)\mathrm{U}(\mathrm{B}\cap \mathrm{C})$$

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