Sure! Let's analyze and solve the given logical expressions step by step. We will be working with the variables A, B, C, D, and E which have the values A = 1, B = 0, C = 1, D = 0, and E = 1.
### 1. ((A AND B) NAND (C NOR D)) AND E
First, let's break it down:
- A AND B: 1 AND 0 = 0
- C NOR D: NOT (C OR D) = NOT (1 OR 0) = NOT 1 = 0
- (A AND B) NAND (C NOR D): NOT ( (A AND B) AND (C NOR D) ) = NOT (0 AND 0) = NOT 0 = 1
- ((A AND B) NAND (C NOR D)) AND E: 1 AND 1 = 1
The result of the first expression is 1.
### 2. ((A NAND C) NAND (B AND D)) OR E
Breaking this down:
- A NAND C: NOT (A AND C) = NOT (1 AND 1) = NOT 1 = 0
- B AND D: 0 AND 0 = 0
- (A NAND C) NAND (B AND D): NOT ( (A NAND C) AND (B AND D) ) = NOT (0 AND 0) = NOT 0 = 1
- ((A NAND C) NAND (B AND D)) OR E: 1 OR 1 = 1
The result of the second expression is True.
### 3. ((A OR B) AND (NOT C NOR NOT D)) AND E
Let's break this down as well:
- A OR B: 1 OR 0 = 1
- NOT C NOR NOT D: NOT (NOT C OR NOT D) = NOT (NOT 1 OR NOT 0) = NOT (0 OR 1) = NOT 1 = 0
- (A OR B) AND (NOT C NOR NOT D): 1 AND 0 = 0
- ((A OR B) AND (NOT C NOR NOT D)) AND E: 0 AND 1 = 0
The result of the third expression is False.
### 4. ((NOT A NOR B) OR (C AND NOT D)) OR NOT E
Breaking it down:
- NOT A NOR B: NOT (NOT A OR B) = NOT (NOT 1 OR 0) = NOT (0 OR 0) = 1
- C AND NOT D: 1 AND NOT 0 = 1 AND 1 = 1
- (NOT A NOR B) OR (C AND NOT D): 1 OR 1 = 1
- ((NOT A NOR B) OR (C AND NOT D)) OR NOT E: 1 OR NOT 1 = 1 OR 0 = 1
The result of the fourth expression is True.
### Final Results
Collecting all these values into the table:
[tex]\[
\begin{tabular}{|c|l|l|l|l|}
\hline & $0-4$ & $5-6$ & $7-8$ & $9-10$ \\
\hline Gate 1 & 1 & & & \\
\hline Gate 2 & True & & & \\
\hline Gate 3 & False & & & \\
\hline Gate 4 & True & & & \\
\hline Total & & & & \\
\hline
\end{tabular}
\][/tex]
Thus, we have:
- Gate 1: 1
- Gate 2: True
- Gate 3: False
- Gate 4: True
Note that True is equivalent to 1 and False to 0 in binary logic.
