Final answer:
To prove the consensus theorem using a truth table, construct a truth table for both sides of the equation and compare their results. If the values are equal for all combinations, the theorem is proven.
Step-by-step explanation:
To prove the consensus theorem using a truth table, we need to construct a truth table for both sides of the equation and compare their results. The theorem states that (B + C) ⋅ (B'+ D) ⋅ (C+ D) is equivalent to (B +C) ⋅ (B'+ D).
Truth Table:
Create a truth table with the variables B, C, D representing the inputs. Assign all possible combinations of true and false values for the variables. Calculate the values for both sides of the equation and compare them. If the values are equal for all combinations, the theorem is proven.
For example, if B = true, C = true, and D = false:
(B + C) ⋅ (B'+ D) ⋅ (C+ D) = (true + true) ⋅ (false' + false) ⋅ (true+ false)
= (true + true) ⋅ (true + false) ⋅ (true + false)
= true ⋅ true ⋅ true = true
(B +C) ⋅ (B'+ D) = (true + true) ⋅ (false + false)
= true ⋅ false = false
Since the values are not equal for this combination, the theorem is not proven.