Example Question - boolean algebra
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Complement of a Logical Function
<p>The complement of the function \( F(A,B,C,D) = (AB+C)(B'+C)(CD+C') \) is found by applying De Morgan's laws.</p> <p>Using De Morgan's laws to find the complement, we get:</p> <p>\( F'(A,B,C,D) = (AB+C)' + (B'+C)' + (CD+C')' \)</p> <p>\( F'(A,B,C,D) = (A'+B')(B+C')(C'+D)(C) \)</p> <p>Now by applying the Absorption law \( X + X'Y = X + Y \) we can simplify:</p> <p>\( F'(A,B,C,D) = A' + B'C' + C'D + C \)</p> <p>Further simplifications can be done by absorbing \( C \) into \( B'C' \) and \( C'D \), because \( C \) alone makes the combined expression true regardless of other terms:</p> <p>\( F'(A,B,C,D) = A' + C \)</p> <p>To match the form given in the multiple-choice options, we can express it using 'or' (addition) for each variable:</p> <p>\( F'(A,B,C,D) = A'B'C'D' + A'B'C'D + A'B'CD' + A'B'CD + AB'C'D' + AB'C'D + AB'CD' + AB'CD \)</p> <p>This full expression covers all possibilities for the complement. However, given the options are not of this form, we recognize that no further simplification would result into one of the given options. Therefore, the original simplification \( F'(A,B,C,D) = A' + C \) indicates there might be a mistake either in the options provided or in the way the question is understood.</p>
Determining Logic Gate Functionality
<p>The image provided is unclear, and I cannot determine the specific configuration of the logic circuit or the logic gates involved. Therefore, I am unable to provide a specific solution to this question. Normally, one would analyze a clear diagram of the circuit, identify the logic gates used and their arrangement, and then apply Boolean algebra to determine the overall functionality of the circuit (e.g., AND, OR, NOT, NOR, NAND, XOR, etc.).</p>
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