QuestionSimplify the Boolean expression: $y=ABC+A\bar{B}(\overline{\bar{A}\bar{C}})$

Studdy Solution

STEP 1

Assumptions1. The expression is a Boolean logic expression. The expression is $y=ABC+A\bar{B}(\overline{\bar{A}\bar{C}})$
3. The bar over a variable represents the NOT operation4. The brackets represent the order of operations5. The plus sign represents the OR operation6. The absence of a sign between variables represents the AND operation

STEP 2

We can simplify the expression by applying De Morgan's law to the term $\overline{\bar{A}\bar{C}}$ . De Morgan's law states that the complement of the conjunction of two variables is equal to the disjunction of their individual complements. In other words, $\overline{A \cdot B} = \bar{A} + \bar{B}$ .
$y=ABC+A\bar{B}(\overline{\bar{A}\bar{C}})$ $y=ABC+A\bar{B}(A+C)$

STEP 3

Next, we can distribute $A\bar{B}$ across $A+C$ using the distributive law of Boolean algebra, which states that $A(B + C) = AB + AC$ .
$y=ABC+A\bar{B}A+A\bar{B}C$

STEP 4

We can simplify the term $A\bar{B}A$ by applying the idempotent law, which states that $A \cdot A = A$ .
$y=ABC+A\bar{B}+A\bar{B}C$

STEP 5

We can rearrange the terms to make the expression clearer.
$y=ABC+A\bar{B}C+A\bar{B}$

STEP 6

We can factor out common terms from the first two terms.
$y=A(BC+\bar{B}C)+A\bar{B}$

STEP 7

We can simplify the term $BC+\bar{B}C$ by applying the consensus theorem, which states that $AB + \bar{A}C + BC = AB + \bar{A}C$ .
$y=A+A\bar{B}$

STEP 8

We can simplify the expression further by applying the absorption law, which states that $A + AB = A$ .
$y=A$ So, the simplified Boolean logic expression is $y=A$ .

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