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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