CSC270 Homework 3 2012

--D. Thiebaut 14:04, 15 February 2012 (EST)

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This assignment is due on 2/22/12. You can work on this assignment in groups of two if you desire.

Problem #1

Draw a 4x4 Karnaugh map of four variables a, b, c, and d, and show several examples of xor or not-xor patterns. Explain what simple rules one should follow to quickly pick out such xor or not-xor patterns in Karnaugh maps.

Problem #2

Use Karnaugh maps to help you find the simplest boolean expression of the following functions. In addition, we know that for the function h (and h only), a can never be equal to b, and c can never be equal to d.

Make sure that if you can simplify (use fewer gates) the result given to you by the Karnaugh map, then you should do so!

• f(a, b, c, d) = Σ( 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15 )

• g(a, b, c, d) = Σ( 0, 1, 2, 3, 12, 13, 14, 15 )

• h(a, b, c, d) = Π( 2, 3, 6, 8, 10, 11, 12, 14, 15 )

Problem #3

Question 1

Assume that you have available to you an active-high output, 4-to-16 decoder, with an active-high enable. Use it along with as few other gates as possible (and, or, not, nand, nor, or xor) to implement the function g below:

g(a, b, c, d) = Σ( 0, 1, 2, 3, 12, 13, 14, 15 )

Question 2

Same as Question 1, but assume that you have a 4-to-1 multiplexer. Find the simplest design (fewer gates) that uses the multiplexer at its core.

Question 3

Same as Question 2, but assume that we now know that a and b are never 0 at the same time, and c and d are never 1 at the same time.