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prove-A-B-C-A-B-C-




Question Number 145459 by syamilkamil1 last updated on 05/Jul/21
prove (A − B)−C = A −(B ∪ C)
$${prove}\:\left({A}\:−\:{B}\right)−{C}\:=\:{A}\:−\left({B}\:\cup\:{C}\right) \\ $$
Answered by liberty last updated on 05/Jul/21
 A−B=A∩B^c   ⇒(A−B)−C = (A∩B^c )∩C^c                                  = A∩(B^c ∩C^c )                                = A∩(B∪C)^c                                = A−(B∪C)
$$\:{A}−{B}={A}\cap{B}^{{c}} \\ $$$$\Rightarrow\left({A}−{B}\right)−{C}\:=\:\left({A}\cap{B}^{{c}} \right)\cap{C}^{{c}} \: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{A}\cap\left({B}^{{c}} \cap{C}^{{c}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{A}\cap\left({B}\cup{C}\right)^{{c}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{A}−\left({B}\cup{C}\right)\: \\ $$
Commented by syamilkamil1 last updated on 07/Jul/21
thanks Sir
$${thanks}\:{Sir} \\ $$

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