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# Homework 1 : A three-input circuit

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Homework 1

Figure 1: A three-input circuit
Figure 2: A three-variable function
Problem 1 Use algebraic manipulation to find
the minimum sum-of-products expression for the
function f = x1x3+x1x¯2+ ¯x1x2x3+ ¯x1x¯2x¯3. [1,
Prob 2.12][10 marks]
Problem 2 Use algebraic manipulation to find
the minimum sum-of-products expression for the
function f = x1x¯2x¯3 + x1x2x4 + x1x¯2x3x¯4. [1,
Prob 2.13][10 marks]
Problem 3 Draw a timing diagram for the circuit in Figure 1. Show the waveforms that can
be observed on all wires (f, g, h, k, l) in the
circuit.[1, Prob 2.8][10 marks]
Problem 4 Represent the function in Figure 2
in the form of a Venn diagram and find its
minimal sum-of-products form. [1, Prob 2.17][10
marks]
Problem 5 Use algebric manipulation to prove
that (x+y)·(x+ ¯y) = x. [1, Prob 2.2] [10 marks].
Problem 6 Determine whether or not the following expressions are valid, i.e., whether the
left- and right-hand sides represent the same
function. [1, Prob 2.7][10 marks]
1. x1x¯3 + x2x3 + ¯x2x¯3 = (¯x1 + ¯x2 + x3)(x1 +
x2 + ¯x3)(¯x1 + x2 + ¯x3)
2. (x1 + x3)(¯x1 + ¯x2 + ¯x3)(¯x1 + x2) = (x1 +
x2)(x2 + x3)(¯x1 + ¯x3)
Problem 7 Design the simplest sum-ofproducts circuit that implements the function
f(x1, x2, x3) = Pm(3, 4, 6, 7). [1, Prob 2.21][10
marks]
Problem 8 Design the simplest product-ofsums circuit that implements the function
f(x1, x2, x3) = Q
M(0, 2, 5). [1, Prob 2.22][10
marks]
References
[1] S. Brown and Z. Vranesic. Fundamentals of
Digital Logic with Verilog Design: Third Edition. McGraw-Hill Higher Education, 2013.
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Homework 1 : A three-input circuit
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