Advanced Math Which one is a simplified expression for A(A'+B')(B+C)(B+C'+D)

Advanced Math Which one is a simplified expression for A(A'+B')(B+C)(B+C'+D)

Question
Upper Level Math
asked 2021-02-05
Advanced Math
Which one is a simplified expression for A(A'+B')(B+C)(B+C'+D)

Answers (1)

2021-02-06
Step 1
Multiply A with the terms in the first bracket:
\(\displaystyle={\left[{A}{\left({A}'+{B}'\right)}\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)
\(\displaystyle={\left[\forall'+{A}{B}'\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)
Since PP'=0
\(\displaystyle={\left[{0}+{A}{B}'\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)
\(\displaystyle={A}{B}'{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)
Step 2
Multiply AB' with the terms in the first bracket of the obtained expression:
\(\displaystyle={\left[{A}{B}'{\left({B}+{C}\right)}\right]}{\left({B}+{C}'+{D}\right)}\)
\(\displaystyle={\left[{A}{B}{B}'+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}\)
\(\displaystyle={\left[{A}{\left({B}{B}'\right)}+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}\)
Since PP'=0
\(\displaystyle={\left[{0}+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}\)
\(\displaystyle={A}{B}'{C}{\left({B}+{C}'+{D}\right)}\)
Step 3
Multiply AB'C with all the terms in the bracket of the obtained expression:
=AB'C(B+C'+D)
=ABB'C+AB'CC'+AB'CD
=A(BB')C+AB'(CC')+AB'CD
Since PP'=0
=0+0+AB'CD
=AB'CD
Thus the simplified expression is AB'CD.
0

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