Problem 24 - Discrete Mathematics with Applications

Question

Using only NAND gates, design a combinatorial circuit that receives \(x\) and \(y\) as input signals and outputs: $$x y$$

Step-by-Step Solution

Answer

The combinatorial circuit for the product x * y using only NAND gates can be constructed in the following way: first, connect inputs x and y to the inputs of a NAND gate (G1) to get NOT (x AND y) as the output. Then, connect the output of G1 to both inputs of a second NAND gate (G2); the output of G2 will be the desired x * y since the double negation cancels out.

Step 1: Understanding NAND gate functionality

A NAND gate is a digital gate that implements logical negation and conjunction. It has two inputs and one output. The truth table for a NAND gate is given below: | Input A | Input B | Output F | |---------|---------|----------| | 0 | 0 | 1 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 0 | From the truth table, we can see that the output F is true when at least one of the inputs is false. In other words, F = NOT (A AND B).

Step 2: Expressing x * y using only NAND gates

Now we need to find a way to express the product x * y using only NAND gates. We know that x * y = x AND y. If we negate both sides, we get: NOT (x * y) = NOT (x AND y) From step 1, we know that the output of a NAND gate is NOT (A AND B), so we can use a NAND gate G1 for this expression with inputs x and y.

Step 3: Double negate the expression

Since we want the output to be x * y and not NOT (x * y), we need to negate our expression once more. Let's use a NAND gate for this operation as well. To do this, we will connect both inputs of a NAND gate G2 to the output of G1: NOT (NOT (x * y)) = NOT (NOT (x AND y)) Since the double negation cancels out, we are left with: x * y = x AND y

Step 4: Construct the combinatorial circuit

Finally, we construct the combinatorial circuit using two NAND gates, G1 and G2: 1. Connect x and y to the inputs of G1. 2. Connect the output of G1 to both inputs of G2. 3. The output of G2 will be the desired product x * y. In summary, the circuit can be constructed using only NAND gates by first using a NAND gate G1 to find NOT (x AND y) and then using a second NAND gate G2 to negate the output of G1, resulting in the desired output x * y.

Key Concepts

NAND Gate

The NAND gate is a fundamental building block in digital electronics. This type of gate performs what is known as a 'universal' function, meaning that any other logic gate can be constructed using just NAND gates. A NAND gate has the unique property of providing an output of 0 only when all its inputs are 1; in all other cases, the output is 1. The behavior of a NAND gate is expressed by its truth table, and with two inputs, the table is as follows:

• If both inputs are 0, the output is 1.
• If one input is 0 and the other is 1, the output is 1.
• If both inputs are 1, the output is 0.

These characteristics make the NAND gate very versatile in digital circuit design. By connecting the outputs and inputs in clever ways, designers can create more complex logic functions, which is especially useful when there's a constraint to use only NAND gates in a design.

Logical Negation and Conjunction

Understanding logical negation and conjunction is crucial in grasping how NAND gates operate. Logical negation, represented by the 'NOT' operator, simply inverts the value of a Boolean input: NOT(0) equals 1, and NOT(1) equals 0. Conjunction, symbolized by the 'AND' operator, checks if both inputs are true (where 1 is true, and 0 is false): if both inputs are 1, AND(1,1) equals 1, otherwise it yields 0 for any other combination of inputs.The NAND gate essentially combines these two operations: it first takes the inputs and applies the AND function, then negates the result. In symbolic form, if A and B are inputs, the output F is thus obtained by NOT(AND(A, B)). An important concept to note is that the NOT operation is idempotent, meaning NOT(NOT(X)) equals X, which allows us to simplify expressions when designing circuits with NAND gates.

Digital Logic Gates

Digital logic gates are the building blocks of digital circuits. These gates process one or more binary inputs to produce a binary output based on logical functions. The most basic types include AND, OR, NOT, NAND, NOR, XOR, and XNOR. Each gate type corresponds to a basic logical operation often used in the algorithms within digital devices like computers, calculators, and many other electronic systems.
For instance:

• AND gates output true only when all inputs are true.
• OR gates output true when at least one input is true.
• NOT gates invert the given input.

Gates can be combined in various ways to create more complex circuits which can perform a range of functions from arithmetic to memory storage. The real beauty of logic gates lies in their ability to be put together, almost like Lego, to build intricate electronic systems that can be found in everything from simple toys to advanced supercomputers.