Identify the Boolean expression corresponding to the truth table given below.
Understand the Problem
The question is asking to identify the correct Boolean expression that corresponds to a given truth table. The truth table shows the values of A, B, and the expected output Y based on those values. We need to analyze the table and determine which of the provided Boolean expressions accurately represents the output Y.
Answer
The Boolean expression is \( (A + B)' \).
Answer for screen readers
The correct Boolean expression is ( (A + B)' ).
Steps to Solve
- List the Truth Table Outputs Identify the given outputs from the truth table.
- For ( A = 0, B = 0 ): ( Y = 1 )
- For ( A = 0, B = 1 ): ( Y = 0 )
- For ( A = 1, B = 0 ): ( Y = 0 )
- For ( A = 1, B = 1 ): ( Y = 0 )
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Determine the Logical Conditions for Y to be 1 From the truth table, ( Y ) is only 1 when both ( A ) and ( B ) are 0. This indicates that ( Y ) is true for the condition: $$ \text{When } A = 0 \text{ AND } B = 0 $$
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Translate Logical Conditions into Boolean Expression The condition where ( Y = 1 ) can be expressed as: $$ Y = \overline{A} \cdot \overline{B} $$
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Identify Equivalent Standard Form The expression ( \overline{A} \cdot \overline{B} ) can be transformed using De Morgan's theorem: $$ Y = \overline{(A + B)} $$
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Match Boolean Expression with Options Provided Now, look through the given options:
- Option 1: ( AB ) - not correct
- Option 2: ( A + B ) - not correct
- Option 3: ( (A + B)' ) - matches
- Option 4: ( A' + B' ) - not correct
- Option 5: None - not correct
Thus, the matching option is ( (A + B)' ).
The correct Boolean expression is ( (A + B)' ).
More Information
This solution shows that ( Y ) is true when both inputs are false. The derived expression ( (A + B)' ) accurately matches the only condition from the truth table where ( Y = 1 ).
Tips
- A common mistake is misinterpreting the conditions under which ( Y ) is true. Ensure to check each possible combination of ( A ) and ( B ) correctly.
- Another mistake can occur when using incorrect logical operations; mixing AND and OR operations can lead to wrong expressions.
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