Sure, let’s go through the process of finding the dual of the given Boolean expression step-by-step.
### Original Boolean Expression
The given Boolean expression is:
[tex]\[ X \bar{Y} Z + X Y Z. \][/tex]
### Step 1: Understand the Dual of a Boolean Expression
The dual of a Boolean expression is obtained by:
1. Replacing all AND operations (denoted by multiplication or ∧) with OR operations (denoted by + or ∨).
2. Replacing all OR operations (denoted by + or ∨) with AND operations (denoted by multiplication or ∧).
3. Keeping the variables and their complements (Not) as they are.
### Step 2: Identify AND and OR operations
In the given expression:
[tex]\[ X \bar{Y} Z + X Y Z, \][/tex]
we can see:
- [tex]\( X \bar{Y} Z \)[/tex] and [tex]\( X Y Z \)[/tex] are terms combined with an OR operation (+).
- Inside each term, there are AND operations since we multiply the variables.
### Step 3: Apply the Dual Operation
To find the dual:
1. Replace the OR operation between the terms with an AND operation:
- The OR operation (+) between [tex]\( X \bar{Y} Z \)[/tex] and [tex]\( X Y Z \)[/tex] becomes an AND operation (∧).
2. Replace the AND operations within each term with OR operations:
- Inside [tex]\( X \bar{Y} Z \)[/tex], the AND operation (.) between [tex]\( X \)[/tex], [tex]\( \bar{Y} \)[/tex], and [tex]\( Z \)[/tex] becomes OR ( ∨ ).
- Inside [tex]\( X Y Z \)[/tex], the AND operation (.) between [tex]\( X \)[/tex], [tex]\( Y \)[/tex], and [tex]\( Z \)[/tex] becomes OR ( ∨ ).
Now, applying these changes yields:
[tex]\[ (X ∨ \bar{Y} ∨ Z) \wedge (X ∨ Y ∨ Z). \][/tex]
### Final Dual Expression
The dual of the given Boolean expression [tex]\( X \bar{Y} Z + X Y Z \)[/tex] is:
[tex]\[ (X ∨ \bar{Y} ∨ Z) \wedge (X ∨ Y ∨ Z). \][/tex]
So, the dual is:
[tex]\[ (X + \not{Y} + Z) \cdot (X + Y + Z). \][/tex]
