Carry Out

CARRY OUT

There are 10 types of people in this world�
those who know binary, and those who don't.

Loo and LooLoo were very satisfied with their little two digit adder, but they were ambitious and wanted their adder to do more than just add two digits together. So they designed a system that output the carry out digit. As in their previous project, they first made a table of all possibile combinations. Their table shows when the sum of three digits creates a carry out digit, and when it does not. They let C_in represent the carry in digit, and C_out represent the carry out digit.


Cin     0   0   0   0   1   1   1   1 
A       0   0   1   1   0   0   1   1
B       0   1   0   1   0   1   0   1
��������������������������������������  
Cout    0   0   0   1   0   1   1   1

Turning their table sideways, presents us with a familiar site. Many of you made a table just like it for your 3 prisoner puzzle. Notice what it says in English: Either A and B are switched on, or C and B are switched on, or C and A are switched on, or C A B are all switched on.

 
    C A B | C_out
    ___________

0)  0 0 0 |  0
1)  0 0 1 |  0
2)  0 1 0 |  0
3)  0 1 1 |  1
4)  1 0 0 |  0
5)  1 0 1 |  1
6)  1 1 0 |  1
7)  1 1 1 |  1
    ___________

Can you design a system using the Pipes applet that implements Loo's and LooLoo's truth table? Of course you can!

HOW TO CONVERT A TRUTH TABLE TO A PIPE GRAM

In those rows, and only those rows, whose output is 1,
AND together the corresponding inputs that equal 1
with the NEGATION of those inputs that equal 0.
You will then have one statement for each row that outputs a 1.
OR together all the statements obtained in step 1.
Applying the equalities you learned in previous assignments, reduce your compound statement from step 2 so that it contains as few & + ~ as possible.
Hint: Review Set Properties.
Test your compound statement with Brian's Truth Table applet.
Diagram your reduced statement. For each & + ~ in your statement, you will need an AND OR NOT block in your pipe diagram.

ASSIGNMENT

For 2 points write an expression for the CAB table using AND OR NOT symbols.
For 3 more points,
draw your expression in the Pipes applet using AND OR NOT blocks.
If you use only 5 blocks, you earn 2 points extra.
If you use only 4 blocks, you earn another 2 points extra.
Copy and save it as a gif file.
Then send it to me privately as an attachment in a message.

Click here for a Carry Out Demo.

Just in case you are curious, a carry out table with ten digits could look like this:

 
   0 1 2 3 4 5 6 7 8 9
�����������������������
0| 0 0 0 0 0 0 0 0 0 0
1| 0 0 0 0 0 0 0 0 0 1
2| 0 0 0 0 0 0 0 0 1 1
3| 0 0 0 0 0 0 0 1 1 1
4| 0 0 0 0 0 0 1 1 1 1
5| 0 0 0 0 0 1 1 1 1 1
6| 0 0 0 0 1 1 1 1 1 1
7| 0 0 0 1 1 1 1 1 1 1
8| 0 0 1 1 1 1 1 1 1 1
9| 0 1 1 1 1 1 1 1 1 1

The 0's represent no carry out digit when adding the row and column digits; the 1's represent a carry out.
For example: column 8 and row 5 indicate a carry out is necesary whenever 8 and 5 are added together:
28 + 35 = 63.

Last Modified 10/10/06 9:41 PM

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