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Boolean Algebra Applications

Boolean algebra can be applied to any system in which each variable has two states. This chapter closes with sample problems solved by

Boolean algebra.

EXAMPLE1

Coffee, Tea, or Milk?

Snerdley's Automated Cafeteria orders a machine to dispense coffee, tea, and milk. Design the machine so that it has a button (input line) for each choice and so that a customer can have at most oneof the three choices. Diagram the circuit to insure that the "at most one" condition is met.

Solution

Step 1Specify the input and output variables and the two states of each. Input c= coffee button (1 = pushed, 0 = not pushed) t= tea button (1 = pushed, 0 = not pushed) m= milk button (1 = pushed, 0 = not pushed) Output x= choice verifier (1 = acceptable input-deliver the choice selected; 0 = unacceptable input-light an error light) Step 2Construct the truth table giving the output desired for each input. xis 1 when exactly oneof c, t, and mis 1. ctmx 1110
1100
1010
1001
0110
0101
0011 0000 Step 3Write a Boolean expression with a term for each 1 output row of the table. ct'm'+ c'tm'+ c't'm

Step 4Try to simplify the formula.

The map shows that the expression cannot be simplified.

Step 5Draw the circuit (next page).

Boolean Algebra Applications1

EXAMPLE2

U.S. Rocket Launcher

The nation of Upper Slobovia has gained a missile defense capability governed by its Security Council. The Council consists of four mem- bers: the U.S. (Upper Slobovian) President and three Counselors (the Chiefs of Staff of the Army and Air Force plus the President's Uncle Homer). The missile system is to be activated by a device obeying these rules: each member of the Security Council has a button to push; the missiles fire only if the President and at least one Counselor push their buttons. Design the rocket firing circuitry.

Solution

Step 1Specify the input and output variables and the two states of each. Input p= President's button (1 = pushed, 0 = not pushed) x, y, z=Counselors' buttons (1 = pushed, 0 = not pushed) Output f= fire missiles command (1 = fire, 0 = don't fire) Step 2Construct the truth table listing all possibilities.

The President and at

least one Counselor agree: fire the mis- siles!

Only the President

pushed his button: do not fire.

The President did

not push his button: do not fire. pxyzf 11111
11101
11011
11001
10111
10101
10011
10000
01110
01100
01010
01000
00110
00100
00010 00000 pxyz pxyz' pxy'z pxy'z' px'yz px'yz' px'y'z

Step 3Write a Boolean expression.

pxyz+ pxyz'+ pxy'z+ pxy'z'+ px'yz+ px'yz'+ px'y'z

Boolean Algebra Applications2

Step 4Simplify the formula.

The map shows three groups of four 1's each, giving this expression. px+ py+ pzor p(x+ y+ z)

Step 5Draw the circuit.

EXAMPLE3

Two-Floor Elevator

Numerous functions must be performed by the circuitry of an elevator (open/close door, move up/down, light up/down indicator, and so on). This example focuses on one aspect of a two-floor elevator: deciding when to move to the other floor.

Solution

Step 1Three inputs are needed.

f= first floor button (1 = pushed, 0 = not pushed) s= second floor button (1 = pushed, 0 = not pushed) p= present floor indicator (1 = 1st floor, 0 = 2nd floor) For f, as the diagram above illustrates, there are two buttons: one out- side the elevator on the first floor and one inside the elevator. Assume that these two buttons are connected in parallel to one line into the system. Similarly for s, assume the "2" buttons inside the elevator and outside the elevator on the second floor are connected. Output m= "move" function (1 = move or change floor; 0 = stay)

Step 2The truth table is on the next page.

The first two input combinations mean that both buttons have been pushed. So stay on the same floor and load passengers there first (that is, m= 0).

Boolean Algebra Applications3

Elevator is on the first floor

and the 2nd floor button is pushed-move. fspm 1110
1100
1010
1001
0111
0100
0010 0000

Elevator is on the 2nd

floor and the 1st floor button is pushed; so move. Step 3The Boolean expression is this: fs'p'+ f'sp Step 4This expression cannot be simplified since the two terms have no com- mon variable.

Step 5Here is the circuit.

EXERCISES 4-8

A

Give all steps in the solution of each problem.

1.Revise Example 1so that the machine offers at most one of fourchoices (add

hot chocolate).

2.Revise Example 2so that there are only two Counselors (eliminate Uncle

Homer). Now design the missile launching system so that it is activated when at least two (any two) of the Council vote to fire.

3.Revise Example 2so that the missiles are launched even without the Presi-

dent's vote if all three Counselors agree to fire.

4.Revise Example 2so that the missiles are launched only if the President and

at least twoof the Counselors agree to fire.

5.For the two-floor elevator of Example 3, design the circuit

that controls the door-opening mechanism. The inputs are the same as in Example 3; replace output mwith output d for opening the door when the elevator is at a floor. C

6.As in Example 3design the circuitry for the "move" func-

tion but now for a three-floor elevator. There are three but- tons: f= first floor button, s= second floor button, and t= third floor button. The present floor indicator requires two bits (xand y): 01 for first floor, 10 for second floor, and 11 for third floor. (Note that 00 is an impossible combination and should be omitted from the table.) For the output m, 1 means move to another floor; 0 means stay. On the next page is a five-variable Karnaugh map.

Boolean Algebra Applications4

7.Most calculators, digital clocks, and watches use the "seven

segment display" format. In this setup, as the diagram at the right shows, there are seven segments that can be lit in differ- ent combinations to form the numerals 0 through 9. For exam- ple, "1" is formed by lighting segments band c; "2" consists of segments a, b, g, e, and d. "4" is composed of segments b, c, f, and g. Design circuitry to run a seven-segment display for one digit. The input consists of a four-bit digit (where each bit is an input line). The outputs are a, b, c, d, e, f, and gof the seven seg- ment diagram (1 = light the segment, 0 = do not light the seg- ment). From a truth table, write and simplify sevenBoolean expressions. Then draw the seven minimal circuits. Note: There are only tenrows of input in the table corresponding to the digits 0 (0000 two through 9 (1001 two

Boolean Algebra Applications5

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