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Question Number 168858 by Shrinava last updated on 19/Apr/22

Simplify: 1. (A + B)(A + B^(−) ) 2. (A^(−) + B)(A^(−) + B^(−) )

$$\mathrm{Simplify}: \\ $$$$\mathrm{1}.\:\left(\mathrm{A}\:+\:\mathrm{B}\right)\left(\mathrm{A}\:+\:\overline {\mathrm{B}}\right) \\ $$$$\mathrm{2}.\:\left(\overline {\mathrm{A}}\:+\:\mathrm{B}\right)\left(\overline {\mathrm{A}}\:+\:\overline {\mathrm{B}}\right) \\ $$

Answered by Rasheed.Sindhi last updated on 19/Apr/22

1. (A + B)(A + B^(−) ) =AA+AB^(−) +BA+BB^(−) =A+AB^(−) +AB+0 =A+A(B^(−) +B) =A+A∙1 =A+A =A 2. (A^(−) + B)(A^(−) + B^(−) ) =A^(−) A^(−) +A^(−) B^(−) +BA^(−) +BB^(−) =A^(−) +A^(−) (B^(−) +B)+0 =A^(−) +A^(−) (1) =A^(−) +A^(−) =A^(−)

$$\mathrm{1}.\:\left(\mathrm{A}\:+\:\mathrm{B}\right)\left(\mathrm{A}\:+\:\overline {\mathrm{B}}\right) \\ $$$$\:\:\:\:\:=\mathrm{AA}+\mathrm{A}\overline {\mathrm{B}}+\mathrm{BA}+\mathrm{B}\overline {\mathrm{B}} \\ $$$$\:\:\:\:\:=\mathrm{A}+\mathrm{A}\overline {\mathrm{B}}+\mathrm{AB}+\mathrm{0} \\ $$$$\:\:\:\:\:=\mathrm{A}+\mathrm{A}\left(\overline {\mathrm{B}}+\mathrm{B}\right) \\ $$$$\:\:\:\:\:=\mathrm{A}+\mathrm{A}\centerdot\mathrm{1} \\ $$$$\:\:\:\:\:=\mathrm{A}+\mathrm{A} \\ $$$$\:\:\:\:\:=\mathrm{A} \\ $$$$\mathrm{2}.\:\left(\overline {\mathrm{A}}\:+\:\mathrm{B}\right)\left(\overline {\mathrm{A}}\:+\:\overline {\mathrm{B}}\right) \\ $$$$\:\:\:\:\:=\overline {\mathrm{A}}\overline {\mathrm{A}}+\overline {\mathrm{A}}\overline {\mathrm{B}}+\mathrm{B}\overline {\mathrm{A}}+\mathrm{B}\overline {\mathrm{B}} \\ $$$$\:\:\:\:\:=\overline {\mathrm{A}}+\overline {\mathrm{A}}\left(\overline {\mathrm{B}}+\mathrm{B}\right)+\mathrm{0} \\ $$$$\:\:\:\:\:=\overline {\mathrm{A}}+\overline {\mathrm{A}}\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:=\overline {\mathrm{A}}+\overline {\mathrm{A}} \\ $$$$\:\:\:\:\:=\overline {\mathrm{A}} \\ $$

Commented by Shrinava last updated on 19/Apr/22

perfect thankyou sir

$$\mathrm{perfect}\:\mathrm{thankyou}\:\boldsymbol{\mathrm{sir}} \\ $$

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