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Question Number 93567 by oustmuchiya@gmail.com last updated on 13/May/20
Given that A={0,1,3,5} B={1,2,4,7} and C={1,2,3,5,8} prove that (i) (A∩B)∩C = A∩(B∩C) (ii) (A∪B)∪C = A∪(B∪C) (iii) (A∪B)∩C = (A∪C)∪(B∩C) (iv) (A∩C)∪B = (A∪B)∩(C∪B)
$${Given}\:{that}\:{A}=\left\{\mathrm{0},\mathrm{1},\mathrm{3},\mathrm{5}\right\}\:{B}=\left\{\mathrm{1},\mathrm{2},\mathrm{4},\mathrm{7}\right\}\:{and}\:{C}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\}\:{prove}\:{that} \\ $$$$\left(\mathrm{i}\right)\:\left(\mathrm{A}\cap\mathrm{B}\right)\cap\mathrm{C}\:=\:\mathrm{A}\cap\left(\mathrm{B}\cap\mathrm{C}\right) \\ $$$$\left(\mathrm{ii}\right)\:\left(\mathrm{A}\cup\mathrm{B}\right)\cup\mathrm{C}\:=\:\mathrm{A}\cup\left(\mathrm{B}\cup\mathrm{C}\right) \\ $$$$\left(\mathrm{iii}\right)\:\left(\mathrm{A}\cup\mathrm{B}\right)\cap\mathrm{C}\:=\:\left(\mathrm{A}\cup\mathrm{C}\right)\cup\left(\mathrm{B}\cap\mathrm{C}\right) \\ $$$$\left(\mathrm{iv}\right)\:\left(\mathrm{A}\cap\mathrm{C}\right)\cup\mathrm{B}\:=\:\left(\mathrm{A}\cup\mathrm{B}\right)\cap\left(\mathrm{C}\cup\mathrm{B}\right) \\ $$
Commented by Ritu Jain last updated on 13/May/20
( i) (A∩B)∩C = A∩(B∩C) (A∩B)={0,1,3,5}∩{1,2,3,7}={1} LHS= (A∩B)∩C=({1})∩{1,2,3,5,8}={1} B∩C={1,2,4,7}∩{1,2,3,5,8}={1} RHS=A∩(B∩C)={0,1,3,5}∩({1}={1} LHS=RHS
$$\left(\:\mathrm{i}\right)\:\left(\mathrm{A}\cap\mathrm{B}\right)\cap\mathrm{C}\:=\:\mathrm{A}\cap\left(\mathrm{B}\cap\mathrm{C}\right) \\ $$$$\:\:\:\:\:\:\:\left(\mathrm{A}\cap\mathrm{B}\right)=\left\{\mathrm{0},\mathrm{1},\mathrm{3},\mathrm{5}\right\}\cap\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{7}\right\}=\left\{\mathrm{1}\right\} \\ $$$$\mathrm{LHS}=\:\left(\mathrm{A}\cap\mathrm{B}\right)\cap\mathrm{C}=\left(\left\{\mathrm{1}\right\}\right)\cap\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\}=\left\{\mathrm{1}\right\} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{B}\cap\mathrm{C}=\left\{\mathrm{1},\mathrm{2},\mathrm{4},\mathrm{7}\right\}\cap\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\}=\left\{\mathrm{1}\right\} \\ $$$$\:\mathrm{RHS}=\mathrm{A}\cap\left(\mathrm{B}\cap\mathrm{C}\right)=\left\{\mathrm{0},\mathrm{1},\mathrm{3},\mathrm{5}\right\}\cap\left(\left\{\mathrm{1}\right\}=\left\{\mathrm{1}\right\}\right. \\ $$$$\mathrm{LHS}=\mathrm{RHS} \\ $$
Answered by otchereabdullai@gmail.com last updated on 14/May/20
ii.( AUB)= {0,1,2,3,4,5,7} C={1,2,3,5,8} ∴ {AUB}UC={0,1,2,3,4,5,7,8} Also A={0,1,3,5} {BUC}={1,2,3,4,5,7,8} ∴AU(BUC)={0,1,2,3,4,5,7,8} Now since (AUB)UC=AU(BUC) ={0,1,2,3,4,5,7,8} we can conclude that (AUB)UC=AU(BUC). iii. (AUB)∩C=(AUC)U(B∩C) (AUB)={0,1,2,3,4,5,7} C={1,2,3,5,8} ∴( AUB)∩C={0,1,2,3,4,5,7}∩{1,2,3,5,8} ={1,2,3,5} also ( AUC)= {0,1,2,3,5,8} (B∩C)= {1,2} ∴ (AUC)U(B∩C)={0,1,2,3,5,8} ⇒(AUB)∩C≠(AUC)U(B∩C)
$$\mathrm{ii}.\left(\:\mathrm{AUB}\right)=\:\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7}\right\} \\ $$$$\:\:\:\:\:\mathrm{C}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\} \\ $$$$\therefore\:\left\{\mathrm{AUB}\right\}\mathrm{UC}=\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7},\mathrm{8}\right\} \\ $$$$\mathrm{Also} \\ $$$$\mathrm{A}=\left\{\mathrm{0},\mathrm{1},\mathrm{3},\mathrm{5}\right\} \\ $$$$\left\{\mathrm{BUC}\right\}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7},\mathrm{8}\right\} \\ $$$$\therefore\mathrm{AU}\left(\mathrm{BUC}\right)=\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7},\mathrm{8}\right\} \\ $$$$\mathrm{Now}\:\mathrm{since} \\ $$$$\left(\mathrm{AUB}\right)\mathrm{UC}=\mathrm{AU}\left(\mathrm{BUC}\right)\:=\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7},\mathrm{8}\right\} \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{conclude}\:\mathrm{that} \\ $$$$\left(\mathrm{AUB}\right)\mathrm{UC}=\mathrm{AU}\left(\mathrm{BUC}\right). \\ $$$$ \\ $$$$ \\ $$$$\:\mathrm{iii}.\:\left(\mathrm{AUB}\right)\cap\mathrm{C}=\left(\mathrm{AUC}\right)\mathrm{U}\left(\mathrm{B}\cap\mathrm{C}\right) \\ $$$$\left(\mathrm{AUB}\right)=\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7}\right\} \\ $$$$\mathrm{C}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\} \\ $$$$\therefore\left(\:\mathrm{AUB}\right)\cap\mathrm{C}=\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7}\right\}\cap\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5}\right\} \\ $$$$\:\:\:\:\:\mathrm{also} \\ $$$$\:\left(\:\mathrm{AUC}\right)=\:\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\} \\ $$$$\:\:\left(\mathrm{B}\cap\mathrm{C}\right)=\:\:\left\{\mathrm{1},\mathrm{2}\right\} \\ $$$$\therefore\:\left(\mathrm{AUC}\right)\mathrm{U}\left(\mathrm{B}\cap\mathrm{C}\right)=\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\} \\ $$$$\Rightarrow\left(\mathrm{AUB}\right)\cap\mathrm{C}\neq\left(\mathrm{AUC}\right)\mathrm{U}\left(\mathrm{B}\cap\mathrm{C}\right) \\ $$

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