Homework #11

Due Wednesday, October 19th, 2005

 

  1. Perform the arithmetic operations below with binary numbers and with negative numbers in signed-2’s complement representation.  Use seven bits to accommodate each number together with its sign.  In each case, determine (and tell) if there is an overflow by checking the carries into and out of the sign bit position

 

(a)              (+ 42) + ( +33)

(b)             (-42)  + (-33)

(c)              (-42) – (+ 33)

 

 

  1. Design a two-bit countdown counter.  This is a sequential circuit with two flip-flops and one input x.  When x = 0, the state of the flop-flops does not change.  When x = 1,  the state sequence is  11, 10, 01, 00, 11, and so on.  Notice that the steps described below mimic what is in your handout and what we went over in class.

 

    1. Show the state diagram
    2. Show the state table with the current state, the input and the next state
    3. add the flip-flop input conditions (outputs of the combinational circuit) that will cause the current state to change to the next state given the value of x
    4. use Karnaugh maps to determine the flip-flop input functions 
    5. Use D flip-flops and draw the circuit

 

  1. Use mathematical induction to prove that the statements below are true for every positive integer n.
    1. 2 + 4 + 6 + ... + 2n = n(n+1)

 

 

    1. 13 + 23+ ... + n3 = n2(n+1)2

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