Answer to Question #209047 in Discrete Mathematics for Ahmed Ali

Question #209047

Find the simplest form for the following boolean expressions:


1) (A.B'.C')+(A'.B'.C')+(A'.B.C')+(A'.B'.C)


2) (A'.B.C)+(A'.B.C)+(A.B.C')+(A.B'.C')+(A'.B.C')+(A'.B'.C')


3) (A+B+C)(A+B'+C')(A+B+C')(A+B+C')


( ' = Not )


1
Expert's answer
2021-06-22T12:26:08-0400

1) ABC+ABC+ABC+ABC=AB(C+C)+ABC+ABC=AB1+ABC+ABC=AB+ABC+ABC=A(B+BC)+ABC=A((B+B)(B+C))+ABC=A(B+C)+ABC=AB+AC+ABC=B(A+AC)+AC=B(A+A)(A+C)+AC=B(A+C)+AC=AB+BC+ACA \cdot \overline B \cdot \overline C + \overline A \cdot \overline B \cdot \overline C + \overline A \cdot B \cdot \overline C + \overline A \cdot \overline B \cdot C = \overline A \cdot \overline B \left( {\overline C + C} \right) + A \cdot \overline B \cdot \overline C + \overline A \cdot B \cdot \overline C = \overline A \cdot \overline {B \cdot } 1 + A \cdot \overline B \cdot \overline C + \overline A \cdot B \cdot \overline C = \overline A \cdot \overline B + A \cdot \overline B \cdot \overline C + \overline A \cdot B \cdot \overline C = \overline A \left( {\overline B + B \cdot \overline C } \right) + A \cdot \overline B \cdot \overline C = \overline A \left( {\left( {\overline B + B} \right)\left( {\overline B + \overline C } \right)} \right) + A \cdot \overline B \cdot \overline C = \overline A \left( {\overline B + \overline C } \right) + A \cdot \overline B \cdot \overline C = \overline A \cdot \overline B + \overline A \cdot \overline C + A \cdot \overline B \cdot \overline C = \overline B \left( {\overline A + A \cdot \overline C } \right) + \overline A \cdot \overline C = \overline B \left( {\overline A + A} \right)\left( {\overline A + \overline C } \right) + \overline A \cdot \overline C = \overline B \left( {\overline A + \overline C } \right) + \overline A \cdot \overline C = \overline A \cdot \overline B + \overline B \cdot \overline C + \overline A \cdot \overline C

Answer: AB+BC+AC\overline A \cdot \overline B + \overline B \cdot \overline C + \overline A \cdot \overline C

2) ABC+ABC+ABC+ABC+ABC+ABC=ABC+ABC+ABC+ABC+ABC=BC(A+A)+ABC+BC(A+A)=BC+ABC+BC=C(B+B)+ABC=C+ABC=(C+AB)(C+C)=C+AB\overline A \cdot B \cdot C + \overline A \cdot B \cdot C + AB\overline C + A \cdot \overline B \cdot \overline C + \overline A B\overline C + \overline A \cdot \overline B \cdot \overline C = \overline A \cdot B \cdot C + AB\overline C + A \cdot \overline B \cdot \overline C + \overline A B\overline C + \overline A \cdot \overline B \cdot \overline C = \overline B \cdot \overline C \left( {A + \overline A } \right) + \overline A \cdot B \cdot C + B\overline C \left( {A + \overline A } \right) = \overline B \cdot \overline C + \overline A \cdot B \cdot C + B\overline C = \overline C \left( {\overline B + B} \right) + \overline A \cdot B \cdot C = \overline C + \overline A \cdot B \cdot C = \left( {\overline C + \overline A \cdot B} \right)\left( {\overline C + C} \right) = \overline C + \overline A \cdot B

Answer: C+AB\overline C + \overline A \cdot B

3)

(A+B+C)(A+B+C)(A+B+C)(A+B+C)=(A+B+C)(A+B+C)(A+B+C)=(A+B+C)(A+C)BB=(A+B+C)(A+C)0=0\left( {A + B + C} \right)\left( {A + \overline B + \overline C } \right)\left( {A + B + \overline C } \right)\left( {A + B + \overline C } \right) = \left( {A + B + C} \right)\left( {A + \overline B + \overline C } \right)\left( {A + B + \overline C } \right) = \left( {A + B + C} \right)\left( {A + \overline C } \right)\overline B B = \left( {A + B + C} \right)\left( {A + \overline C } \right) \cdot 0 = 0

Answer: 0


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