Answer to Question #217671 in Discrete Mathematics for Kunal

Question #217671

Produce a truth table for given Boolean expression (A+B'+C)(A+B+C)(A'+B+C')


1
Expert's answer
2021-07-26T15:36:27-0400

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


Laws Used:

AA=AA+AB=A  [Absorption Law]A\cdot A =A\\A+AB=A\ \ [\text{Absorption Law}]

AA=0A+0=AA+A=AA+1=1A\cdot A'=0\\A+0=A\\A+A=A\\A+1=1


Now ,

    (A+B+C)(A+B+C)(A+B+C)    (A+AB+AC+AB+BC+AC+BC+C)(A+B+C)    (AA+AB+AC+AAB+ABB+ABC+AAC+ABC+ACC+AAB+ABB+ABC+ABC+BBC+BCC+AAC+ABC+ACC+ABC+BBC+BCC+AC+BC+CC)\implies (A+B'+C)(A+B+C)(A'+B+C')\\\implies (A+AB+AC+AB'+B'C+AC+BC+C)(A'+B+C')\\\implies (AA'+AB+AC'+AA'B+ABB+ABC'+AA'C+ABC+ACC'+AA'B'+ABB'+AB'C'+A'B'C+BB'C+B'CC'+AA'C+ABC+ACC'+A'BC+BBC+BCC'+A'C+BC+CC')


    (0+AB+AC+0+AB+ABC+0+ABC+0+0+0+ABC+ABC+0+0+0+ABC+0+ABC+BC+0+AC+BC+0)    (AB+BC+AC+AC+ABC+ABC+ABC+ABC+ABC)    AB(1+C+C)+BC(1+A)+AC(1+B)+AC(1+B)    AB+BC+AC+AC\implies (0+AB+AC'+0+AB+ABC'+0+ABC+0+0+0+AB'C'+A'B'C+0+0+0+ABC+0+A'BC+BC+0+A'C+BC+0)\\\implies (AB+BC+AC+A'C+ABC+ABC'+AB'C'+A'B'C+A'BC)\\\implies AB(1+C+C')+BC(1+A')+AC'(1+B')+A'C(1+B)\\\implies AB+BC+AC'+A'C

    \impliesAB+A'C+AC'\


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