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Homework Assignment: Consensus Theorem, Associativity Proofs, and Logic Expression Conversions

This homework assignment requires students to prove the dual of the Consensus Theorem (6 marks), validate if given associativity relationships are correct or incorrect (3 marks each), and convert several logic expressions between Sum of Products (SOP) and Product of Sums (POS) forms (3 marks each). Additionally, implement the given function using different configurations of 8-to-1, 4-to-1, and 2-to-1 line multiplexers (3 marks each). Prepare for critical analysis and practical implementation in digital logic circuits.

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Homework Assignment: Consensus Theorem, Associativity Proofs, and Logic Expression Conversions

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  1. Homework 1 Due 21st May 2014

  2. Prove the Dual of the following Consensus Theorem x.y + x’.z + y.z = x.y + x’.z Marks = 6

  3. Prove whether the following Associativity Relationships are correct or incorrect? a ^ (b ^ c) = (a ^ b) ^ c a (b c) = (a b) c a (b c) = (a b) c Note: The legend for the aforementioned symbols is as follows ^ XOR NAND NOR Marks = 3 + 3 + 3

  4. Convert the following SOP expression into a POS expression • f = x.y + x’.y.z’ + y’.z’ Convert the following POS expression into a SOP expression • f = ( x + y ).( x’ + y + z’ ).( y’ + z’ ) Marks = 3 + 3

  5. Convert the following SOP expression into a C-SOP expression • f = x.y + x’.y.z’ + y’.z’ Convert the following POS expression into a C-POS expression • f = ( x + y ).( x’ + y + z’ ).( y’ + z’ ) Marks = 3 + 3

  6. Implement This Function using • two 8-to-1 Line Multiplexers • two 4-to-1 Line Multiplexers • two 2-to-1 Line Multiplexers • two 8-to-1 Line Multiplexer s (where the Multiplexers are tree structures of smaller multiplexers) Marks = 3 + 3 + 3 + 4

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