In digital logic, we often think in terms of three fundamental operations: AND, OR, and NOT. They are the conceptual building blocks for complex circuits. However, when it comes to manufacturing integrated circuits, using a wide variety of gate types is inefficient. It's cheaper and simpler to produce chips using just one type of gate over and over.

This is where universal gates come in. The NAND and NOR gates are special because either one can be used to create any other logic function. This property is called functional completeness.

Functional completeness means a set of logic gates can be used to generate any possible truth table. Since NAND and NOR gates can each replicate the functions of AND, OR, and NOT, they are functionally complete on their own. This allows engineers to design complex logic using a single, standardized component, simplifying the design and manufacturing process significantly.

To prove the NAND gate's universality, we just need to show that we can construct the three basic logic gates using only NAND gates.

Creating an AND Gate: A NAND gate is literally a "Not AND" gate. To get the original AND function, we just need to invert the output of a NAND gate. We can do this by feeding the output of the first NAND gate into a second NAND gate configured as an inverter.

Creating an OR Gate: This construction is a bit more clever and relies on De Morgan's laws. The expression for an OR gate is $A+B$. According to De Morgan's laws, this is equivalent to $\overline{\overline{A} \cdot \overline{B}}$.

This expression tells us exactly how to build the circuit. First, we need NOT A and NOT B. We can get these by using two NAND gates as inverters. Then, we need to AND those inverted signals together and invert the result, which is exactly what a third NAND gate does.

Just like the NAND gate, the NOR gate is also functionally complete. We can prove this by following a similar process of constructing the three fundamental gates using only NOR gates. The principles are analogous, often relying on the other of De Morgan's laws.

Creating an OR Gate: This is the counterpart to building an AND from NANDs. A NOR gate is a "Not OR" gate. To recover the OR function, we simply invert its output using another NOR gate configured as an inverter.

Creating an AND Gate: Again, we turn to De Morgan's laws. The expression for an AND gate is $A \cdot B$. This is equivalent to $\overline{\overline{A} + \overline{B}}$.

The structure mirrors the NAND-based OR gate. We use two NOR gates as inverters to get NOT A and NOT B. Then, we feed these signals into a third NOR gate, which performs the OR operation and the final inversion.

The ability to construct all basic logic from a single gate type is a cornerstone of modern digital design. It allows for optimized, uniform, and cost-effective fabrication of the complex circuits that power our technology.