Logic Gates

Show your work for each problem.

1.
Which gate gives output 1 only when both inputs are 1?
1. a)OR
2. b)XOR
3. c)NOR
4. d)AND
Answer: AND
AND gives 1 only for the inputs 1, 1. OR and XOR already give 1 when just one input is 1, and NOR gives 1 only for 0, 0.
2.
An OR gate has the inputs $A = 0$ and $B = 1$ . What is the output? (Write 1 or 0.)
Answer: 1
$0 \lor 1 = 1$ , because input $B$ is 1.
3.
A NOR gate has the inputs $A = 0$ and $B = 0$ . What is the output? (Write 1 or 0.)
Answer: 1
$0 \lor 0 = 0$ , and NOT flips it: $\overline{0 + 0} = 1$ . NOR gives 1 only when both inputs are 0.
4.
For the inputs $00$ , $01$ , $10$ , $11$ a gate gives the outputs $0, 1, 1, 0$ . Which gate is it?
1. a)XOR
2. b)NAND
3. c)AND
4. d)OR
Answer: XOR
The output is 1 exactly when the inputs are different, which is XOR. OR would give 1 in the last row as well.
5.
The inputs are $A = 1$ and $B = 1$ . Find the output of an AND, an OR and an XOR gate.
Answer: 1, 1, 0
1. What is the output of the AND gate? 1
2. What is the output of the OR gate? 1
3. What is the output of the XOR gate? 0
6.
$A$ goes through a NOT gate, and its output and $B$ go into an OR gate: $Y = \neg A \lor B$ . Find $Y$ for $A = 1$ , $B = 0$ .
Answer: 0
1. What is the output of the NOT gate, $\neg A$ ? 0
2. What is $Y = 0 \lor B$ with $B = 0$ ? 0
7.
$A$ and $B$ go into an OR gate, and its output goes through a NOT gate. Which expression describes the circuit?
1. a) $\neg A \land B$
2. b) $\neg(A \lor B)$
3. c) $\neg A \lor \neg B$
4. d) $\neg(A \land B)$
Answer: $\neg(A \lor B)$
First OR: $A \lor B$ . Then NOT on that output: $\neg(A \lor B)$ , which is the NOR gate $\overline{A + B}$ .
8.
In how many of the 4 input rows does a NAND gate give the output 1?
Answer: 3
AND is 1 only in the row $11$ . NAND flips every row, so it is 1 in the other $4 - 1 = 3$ rows.
9.
$A$ and $B$ go into an AND gate. $B$ and $C$ go into an OR gate. Both outputs go into an XOR gate, which gives $Y$ . Find $Y$ for $A = 1$ , $B = 0$ , $C = 1$ .
Answer: 1
1. What is the output of the AND gate, $A \land B$ ? 0
2. What is the output of the OR gate, $B \lor C$ ? 1
3. What is $Y = 0 \oplus 1$ ? 1
10.
A circuit computes $Y = (A \land B) \lor \neg C$ . Find $Y$ for $A = 0$ , $B = 1$ , $C = 1$ .
Answer: 0
1. What is $A \land B$ ? 0
2. What is $\neg C$ ? 0
3. What is $Y$ , the OR of the two results? 0
11.
In gate notation ( $\cdot$ = AND, $+$ = OR, bar = NOT), find $Y = \overline{A \cdot B} + C$ for $A = 1$ , $B = 1$ , $C = 0$ .
Answer: 0
$A \cdot B = 1$ , the bar flips it to $0$ , and $0 + 0 = 0$ (OR). So $Y = 0$ .
12.
Which inputs make a NOR gate give the output 1?
1. a)A = 1, B = 1
2. b)A = 1, B = 0
3. c)A = 0, B = 0
4. d)A = 0, B = 1
Answer: A = 0, B = 0
OR is 0 only for $A = 0$ , $B = 0$ , so NOR $= \overline{A + B}$ is 1 only for those inputs.
13.
A half adder gets $A = 0$ and $B = 1$ . It outputs the sum digit $S = A \oplus B$ and the carry digit $C = A \land B$ . Find $S$ , $C$ and the value of the binary number $CS$ .
Answer: 1
1. What is $S = 0 \oplus 1$ ? 1
2. What is $C = 0 \land 1$ ? 0
3. The result is $CS = 01$ in binary. Which ordinary number is that? 1
14.
A circuit computes $Y = (A \oplus B) \land \neg C$ . In how many of the 8 input combinations is $Y = 1$ ?
Answer: 2
1. For how many of the 4 pairs $AB$ is $A \oplus B = 1$ ? 2
2. For how many values of $C$ is $\neg C = 1$ ? 1
3. How many of the 8 combinations give Y = 1? 2
15.
Both inputs of a NAND gate are connected to the same signal $A$ . Find the output for $A = 0$ and for $A = 1$ , and decide how many NAND gates you need to build one AND gate.
Answer: 2
1. What is the output for $A = 0$ , that is $\overline{0 \cdot 0}$ ? 1
2. What is the output for $A = 1$ , that is $\overline{1 \cdot 1}$ ? 0
3. AND is a NAND followed by a NOT. How many NAND gates is that? 2
16.
Which circuit gives the same output as an XOR gate for all inputs?
1. a) $(A \lor B) \land \neg(A \land B)$
2. b) $(A \land B) \lor \neg(A \lor B)$
3. c) $\neg A \land \neg B$
4. d) $(A \lor B) \land (A \land B)$
Answer: $(A \lor B) \land \neg(A \land B)$
$(A \lor B) \land \neg(A \land B)$ gives $0, 1, 1, 0$ for $00, 01, 10, 11$ , the XOR column. The second option gives $1, 0, 0, 1$ , the third is NOR and the fourth is just AND.
17.
Three judges press a button (1) or not (0). A light shows $Y = (A \land B) \lor (A \land C) \lor (B \land C)$ . Find $Y$ for $A = 1$ , $B = 0$ , $C = 1$ , then count the input combinations with $Y = 1$ .
Answer: 4
1. What is $A \land B$ ? 0
2. What is $A \land C$ ? 1
3. What is $B \land C$ ? 0
4. What is $Y$ ? 1
5. In how many of the 8 combinations is Y = 1? 4
18.
A circuit computes $Y = \overline{A + B} + A \cdot B$ (bar = NOT, $+$ = OR, $\cdot$ = AND). In how many of the 4 input rows is $Y = 1$ ?
Answer: 2
Row $00$ : $\overline{0} + 0 = 1$ . Rows $01$ and $10$ : $\overline{1} + 0 = 0$ . Row $11$ : $\overline{1} + 1 = 1$ . So $Y = 1$ in 2 rows: exactly when the inputs are equal, the opposite of XOR.