MHB Analysis. Fill in the F column values in the truth table for the circuit.

AI Thread Summary
The discussion revolves around filling in the F column values of a truth table for a circuit and checking the accuracy of the calculations. The initial response included a truth table with values for x, y, z, and their corresponding outputs for logical operations. A key point of confusion was the interpretation of the NAND operation, represented by the Sheffer stroke (x NAND y), which is defined as "not both" and is equivalent to the negation of the AND operation. Participants clarified that the correct approach to compute x NAND y involves first calculating the AND of x and y, then negating the result, rather than negating x and y individually before ANDing. This misunderstanding led to incorrect values in the truth table. The discussion concluded with a participant expressing gratitude for the detailed explanations and indicating they would rework their solution based on the clarified logic.
shamieh
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Fill in the F column values in the truth table for the circuit.
Need someone to check my work.

View attachment 1444

My Answer:
  • x y z | f | x! and y! | x! XOR z|
  • 0 0 0| 0 | 1 | 1
  • 0 0 1| 1 | 1 | 0
  • 0 1 0| 1 | 0 | 1
  • 0 1 1| 0 | 0 | 0
  • 1 0 0| 0 | 0 | 0
  • 1 0 1| 1 | 0 | 1
  • 1 1 0| 1 | 1 | 0
  • 1 1 1| 0 | 1 | 1
 

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So, let's define the $\uparrow$ symbol as the Sheffer stroke, which corresponds to the NAND logic gate you have there. Then $f=(x \uparrow y) \oplus (x \oplus z)$. The truth table is then
$$
\begin{array}{c|c|c|c|c|c}
x &y &z &x \uparrow y &x \oplus z &(x \uparrow y) \oplus (x \oplus z) \\ \hline
0 &0 &0 &1 &0 &1 \\
0 &0 &1 &1 &1 &0 \\
0 &1 &0 &1 &0 &1 \\
0 &1 &1 &1 &1 &0 \\
1 &0 &0 &1 &1 &0 \\
1 &0 &1 &1 &0 &1 \\
1 &1 &0 &0 &1 &1 \\
1 &1 &1 &0 &0 &0
\end{array}
$$
 
Okay, I see where I went wrong. But you're saying x NAND y in the first row is 0 NAND 0 which is really 1 AND 1 = 1. I get that. Then you are saying in the 2nd row the same thing 0 NAND 0 = 1 because it's really 1 AND 1. Then in the 3rd row you are saying 0 NAND 1 which is really 1 AND 0 = 1. But how does that = 1? 1 AND 0 = 0. It only equals 1 if both are 1.
 
shamieh said:
Okay, I see where I went wrong. But you're saying x NAND y in the first row is 0 NAND 0 which is really 1 AND 1 = 1.

Actually not. The NAND operation means "not both". If I say $x \uparrow y$, or $x$ NAND $y$, that is equivalent to $\overline{xy}$. By DeMorgan, $ \overline{xy}= \bar{x}+ \bar{y}$, not $ \underbrace{\overline{xy}= \bar{x} \bar{y}}_{\text{Wrong!}}$.

I get that. Then you are saying in the 2nd row the same thing 0 NAND 0 = 1 because it's really 1 AND 1. Then in the 3rd row you are saying 0 NAND 1 which is really 1 AND 0 = 1. But how does that = 1? 1 AND 0 = 0. It only equals 1 if both are 1.

Again, this reasoning is flawed. If you need to, calculate NAND's like this: to compute $x \uparrow y$, first compute $xy$, and then negate the result. You cannot compute NAND's by negating the $x$ and $y$ first, and then AND'ing the results.
 
Ahh! I see! So DeMorgans law changes the ANDing of xy to OR, while negating the terms as well. x! + y! ok. Going to re-work it and make sure I get the correct solution. I'll be back (Wait)

- - - Updated - - -

Awsome, thank you for the detailed explanations. (Sun)
 
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