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Digital Logic Worked Example: <boolean equations>

Module by: Cathy Huang. E-mail the authorEdited By: Elec and Comp 326

Summary: Boolean equation simplification practice examples

Module: Digital Logic Worked Example: Boolean Equations

Simplify the following Boolean equations.
Sketch combinational circuit implementation for each equation.

Exercise 1

Y=B¬C+BC+AB Y B C B C A B

Solution

Y=B¬C+BC+AB Y B C B C A B
apply Distributivity Thm:
Y=B(¬C+C)+AB Y B C C A B
apply Complements Thm:
Y=B+AB Y B A B
apply Distributivity Thm:
Y=B(1+A) Y B 1 A
apply Null Element Thm:
Y=B Y B

Exercise 2

Y=BA+BC(B+C) Y B A B C B C

Solution

Y=BA+BC(B+C) Y B A B C B C
apply Distributivity:
Y=AB+BBC+BCC Y A B B B C B C C
apply Idempotency
Y=AB+BC+BC Y A B B C B C
apply apply Idempotency
Y=AB+BC Y A B B C
apply Distributivity
Y=B(A+C) Y B A C

Exercise 3

Y=¬A(C(C+B)+¬BC) Y A C C B B C

Solution

Y=¬A(C(C+B)+¬BC) Y A C C B B C
apply Distributivity
Y=¬A(CC+CB+¬BC) Y A C C C B B C
apply Idempotency
Y=¬A(C+CB+C¬B) Y A C C B C B
apply Distributivity
Y=¬A(C+C(B+¬B)) Y A C C B B
apply Complements
Y=¬A(C+C) Y A C C
apply Idempotency
Y=¬AC Y A C

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