Question

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

Upper Level Math
ANSWERED
asked 2021-02-05

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

Expert 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.

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