Matrix Representation of Basic Logic Gates

Represent the following classical logic gates as matrices and explain the logic behind each matrix representation:

NOT Gate

AND Gate

Constructing a Logic Operation Using Basic Gates

Given the matrices representing the NOT and AND gates from Problem $1,$ construct a matrix representation for the NAND gate using matrix operations. Explain how the combination of these operations represents the NAND logic.

$3.$ Implementing an OR Gate Using a Matrix

Design a matrix that represents the OR operation for two inputs. Each input can be $0$ or $1,$ and the output should correctly reflect the OR operation. Explain how your matrix operates on input vectors to produce the correct output. Show the output of your OR gate for each possible input.

$4.$ Understanding the CNOT Gate in a Classical Context

The Controlled$-$NOT $($CNOT$)$ gate is a fundamental quantum gate, but let's conceptualize it in a classical context. The CNOT gate flips the second bit if the first bit is $1.$ Represent this operation as a matrix. Describe how this matrix can be interpreted in a scenario involving classical bits.

$5.$ Combining Logic Gates in a Circuit

Imagine a logical circuit where an AND gate feeds into a CNOT gate, where the output of the AND gate is the control bit for the CNOT. If the inputs of the AND gate are represented by $x$ and $y,$ and an independent input $z$ serves as the target for the CNOT, construct a matrix that represents the entire operation. Explain the process and the resulting matrix.