Answer to Question #120552 in Discrete Mathematics for inv

Question

Find the simplest form for the following boolean expressions using algebraic methods.
1. (A+B̄)(B+C)+(A+B)(C+Å̄)
2. (A+B)(AC+AC̄)+AB+B

Answer 1

1 )

$AB+AC+B.\tilde{B}+\tilde{B}.C+AC+A.\tilde{A}+BC+B.\tilde{A}$

$AB+AC+\tilde{B}.C+AC+BC+B.\tilde{A}$

$AB+AC+\tilde{B}.C+BC+B.\tilde{A}$

$AB+AC+C(\tilde{B}+B)+B.\tilde{A}$

$B(A+\tilde{A})+AC+C$

$B+AC+C$

$B+C(A+1)$

$B+C$

2 )

$(A+B)(A.C+A.\tilde{C})+A.B+B$

SOLUTION:-

=$A.C+A.\tilde{C}+A.B.C+A.B.\tilde{C}+A.B+B$

=$A.C+A.\tilde{C}+A.B.C+A.B.\tilde{C}+A.(B+1)$

=$A.C+A.\tilde{C}+A.B.C+A.B.\tilde{C}+A$

=$A.C+A.\tilde{C}+A.B.(C+\tilde{C})+A$

=$A.C+A.\tilde{C}+A.B+A$

=$A.C+A.\tilde{C}+A.(B+1)$

=$A.C+A.\tilde{C} +A$

=$A.(C+\tilde{C})+A$

=A+A

=A

LAWS:-

1 ) $x.\tilde{X}=0$

2 ) x.x=x

3 ) x+x=x

4 ) x+1=1

5 ) $x+\tilde{x}=1$