Boolean algebra and NAND gate

Discussion in 'Homework Help' started by icelated, Oct 15, 2010.

1. icelated Thread Starter New Member

Oct 15, 2010
12
0
I am a software engineering student taking a digital logic class, and i am trying to multiply this binary number. I am not sure what a carry in is and a Carry out. the teachers slides are horrible. It appears he used a truth table to do this but its confusing.
Code ( (Unknown Language)):
1.
2.    X1X0
3.  + Y1Y0
4.    -----
5.  Z2Z1Z0
6.
7.
I think thats how its set up! Now, for the multiplication part. Oh, i think the carry in and carry out is for adding?
If so, dont worry about the carry in and carry out part ok?
but i think i still need to use the multiplication answer for the table?

Code ( (Unknown Language)):
1.
2.
3.        1  carry in?  like if we are adding? Now, that i think about it there is no carry in with multiplication!
4.    110101
5.    X 1101
6.    ------
7.  101011001  thats what i ended up with. Probably, not right!
8.
9.
I think my truth table should look something like this: keep in mind this is not set up to my answer above

Code ( (Unknown Language)):
1.
2.        X1X0
3.      + Y1Y0
4.         ----
5.      Z2Z1Z0
6.
7.        X0   Y0   Carry     Z0
8.        0     0     0          0
9.        1     0     0          1
10.        0     1     0          1
11.        1     1     1          0
12.
13.
14.
15.   X1    Y1     Carryin               Carryout    Z1
16.   0      0           0                    0             0
17.   1      0           0                    0             1
18.   0      1           0                    0             1
19.   1      1           0                    1             0
20.   0      0           1                    0             1
21.   1      0           1                    1             0
22.
23.
I get confused on the x1 and y1 part It would be easier if i can see it in action and labeled what the "carry in" is and "carry out" is while its being multiplied OR adding?.

would the "carry in" be the result of 1+1 and the "carry out" be the result of the next carry result?
I think after we get the truth table done with the carry in and carry out we are to use boolean algebra like:

Code ( (Unknown Language)):
1.
2.   Z1 = X1• Y1' • Carryin' + X1' • Y1• Carryin' + X1' • Y1' • Carryin + X1• Y1• Carryin
3. Carryout = X1• Y1• Carryin' + X1 • Y1' • Carryin + X1' • Y1• Carryin + X1 • Y1• Carryin
4. Z2 = Carryout
5.
6.
We are to "work out the equations for the AND, OR and NOT functions using only the NAND operator." not sure how to do this!

2. Georacer Moderator

Nov 25, 2009
5,151
1,266
First things first! You must learn how to add binary numbers correctly. In this example we will learn how to add two 4-bit numbers.

The carry-in refers to a possible 1 that comes from a previous addition and must be taken into account for the current addition. Carry out refers to a possible carry that will be produced from our addition, if the result is larger than 4 bits (it can be at most 5 bits). It is a product of the current addition, not an operand

Suppose we want to add the numbers 1101 (12) and 1000 (8) and we have a carry in from a previous addition:

$\begin{array}{ccccc} B4 & B3 & B2 & B1 & B0 \\
\ &\ &\ &\ &1\\
\ &1&1&0&1\\
+&1&0&0&0\\
-&-&-&-&-\\
1&0&1&1&0
\end{array}$

We make the additions of every column, starting from the right. In the result, we write only the last bit of the minor sum.
I remind that 0+0=0, 0+1=1, 1+1=10 and 1+1+1=11.

That said B0=1+1+0=10. We write 0 on the result and keep 1 as a carry for the next step.
B1=1+0+0=1
B2=1+0=1
B3=1+1=10
B4=1

As the result is larger (5 bits) than the operands, we usually say that we have an overflow and say that B4=1 is the carry out, while the result is 0110.

As for mutliplication, it is done exactly as in the decimal system. Refer here: http://en.wikipedia.org/wiki/Binary_multiplication#Multiplication for more info.