Digital Logic - truth tables

Question

I am trying to solve these problems with truth tables using the formulas below. I am having a problem with the NOT to NAND

I think i got the first 2 problems correct using: AND is equivalent to NOR, AND is equivalent to NAND

The equations for AND, OR and NOT using the NAND operator are:

X + Y = x' NAND y' ??
X * Y = 
X' = X NAND 1

The equations for AND, OR and NOT using only the NOR operator are:

X' = ?
X + Y = (X NOR Y) NOR 0
X * Y = X' NOR Y' ??

I am trying to follow this formula

An OR gate with a NOT in both inputs is equivalent to a NAND.
An AND gate with a NOT in both inputs is equivalent to a NOR.
An OR gate with a NOT in its output is equivalent to a NOR.
An AND gate with a NOT in its output is equivalent to a NAND.

this is how i worked out AND to NOR equivalent

X  Y    X AND Y      X NOR Y    
-----------------------------
0  0       0            1
0  1       0            0
1  0       0            0
1  1       1            0

X'  Y'   X' NOR Y'
-------------------
1   1        0
1   0        0
0   1        0
0   0        1

My biggest problem is how do i use a NOT in the outputs in a truth table? Also, how would i go from a NOT to a NOR? I want to work all these out in a truth table.

X * Y = "An AND gate with a NOT in its output is equivalent to a NAND."

what are the outputs in a truth table? and how can i go from a X' to equivalent NAND?

Answer 1

how do i use a NOT in the outputs in a truth table

Put a bar over the column label, if you want to indicate reverse-polarity on the output.

Also, how would i go from a NOT to a NOR

Tie the two inputs of the NOR together. That makes it the equivalent of a NOT.

Answer 2

We know that,

X   X'
------
0   1
1   0 

Hence,

X   X NOR X
------------
0      1
1      0 

So we get,

X' = X NOR X

For your second question, suppose you have two inputs X' & Y' and you want to get to X NAND Y,

X NAND Y = (X AND Y)' = X' OR Y' (by DeMorgan's Law)

Hence, the truth tables are as follows:

X  Y    X AND Y     (X AND Y)'    
-----------------------------
0  0       0            1
0  1       0            1
1  0       0            1
1  1       1            0

X  Y    X'   Y'      X' OR Y'    
-----------------------------
0  0    1    1          1
0  1    1    0          1
1  0    0    1          1
1  1    0    0          0