Question

Use Boolean algebra to show that:
Aˉ⋅B+Cˉ⋅Bˉ+A+C=Aˉ⋅B+Cˉ⋅Bˉ

Answer 1

Using Boolean algebra,
`\(A^\sim \cdot B + C^\sim \cdot B^\sim + A + C = A^\sim \cdot B + C^\sim \cdot B^\sim\).`

Step-by-step explanation:

To demonstrate the given expression using Boolean algebra, let's break it down step by step. Starting with the left-hand side (LHS) of the equation:

`\(A^\sim \cdot B + C^\sim \cdot B^\sim + A + C\)`

Applying the absorption law
`(\(X + X^\sim Y = X + Y\)) t`o the term \(A^\sim \cdot B + A\) results in \(A^\sim + B\). Then, utilizing the identity law (\(X +
`X^\sim Y = X + Y\)) on the term \(C^\sim \cdot B^\sim + C\) yields \(C^\sim + B^\sim\).`

Thus, combining the simplified expressions gives \(
`A^\sim + B + C^\sim + B^\sim\).`By applying the consensus theorem
`(\(XY + X^\sim Z + YZ = XY + X^\sim Z\))`, the equation further simplifies to \(A^\sim \cdot B + C^\sim \cdot B^\sim\), which matches the right-hand side (RHS) of the given expression.

Hence, using various Boolean algebra laws and theorems, we've shown that the expression
`\(A^\sim \cdot B + C^\sim \cdot B^\sim + A + C\) simplifies to \(A^\sim \cdot B + C^\sim \cdot B^\sim\).`