Question

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

Upper Level Math

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

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.