Question

Problem 1. Design a circuit with four inputs and one output so that the output is 1 if there are an odd numbers of ones among the input variables, and 0 otherwise. (Use only AND, OR, NOT gates.) Problem 2. Construct a circuit equivalent to the half-adder (from class and in the textbook) using only NAND gates. Use as few NAND gates as you can. Problem 3. Recall from class that we can write all three of AND, OR, NOT in terms of NAND. We can express this in terms of the following logical equivalences for the Sheffer stroke $P \mid Q$. $\sim P = P \mid P$ $P \lor Q = (P \mid P) \mid (Q \mid Q)$ $P \land Q = (P \mid Q) \mid (P \mid Q)$ Write logical equivalences to express $\sim P$, $P \lor Q$, $P \land Q$ in terms of the Pierce arrow $\downarrow$ which represents the NOR operation. Prove your identities using truth tables.

          Problem 1. Design a circuit with four inputs and one output so that
the output is 1 if there are an odd numbers of ones among the input
variables, and 0 otherwise. (Use only AND, OR, NOT gates.)
Problem 2. Construct a circuit equivalent to the half-adder (from
class and in the textbook) using only NAND gates. Use as few NAND
gates as you can.
Problem 3. Recall from class that we can write all three of AND, OR,
NOT in terms of NAND. We can express this in terms of the following
logical equivalences for the Sheffer stroke $P \mid Q$.
$\sim P = P \mid P$
$P \lor Q = (P \mid P) \mid (Q \mid Q)$
$P \land Q = (P \mid Q) \mid (P \mid Q)$
Write logical equivalences to express $\sim P$, $P \lor Q$, $P \land Q$ in terms of
the Pierce arrow $\downarrow$ which represents the NOR operation. Prove your
identities using truth tables.

Problem 1. Design a circuit with four inputs and one output so that
the output is 1 if there are an odd numbers of ones among the input
variables, and 0 otherwise. (Use only AND, OR, NOT gates.)
Problem 2. Construct a circuit equivalent to the half-adder (from
class and in the textbook) using only NAND gates. Use as few NAND
gates as you can.
Problem 3. Recall from class that we can write all three of AND, OR,
NOT in terms of NAND. We can express this in terms of the following
logical equivalences for the Sheffer stroke P | Q.
∼ P = P | P
P Q = (P | P) | (Q | Q)
P Q = (P | Q) | (P | Q)
Write logical equivalences to express ∼ P, P Q, P Q in terms of
the Pierce arrow ↓ which represents the NOR operation. Prove your
identities using truth tables.

Added by Steven H.

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