State the function, truth table and Boolean expression of the NAND and NOR gates and explain why NAND is universal

What this dot point is asking

SEAB wants you to state the function, truth table and Boolean expression of the NAND and NOR gates, and to explain why the NAND gate is called universal. The central insight is that NAND and NOR are simply AND and OR followed by a NOT, so their outputs are the inverses of the AND and OR outputs, and that NAND is special because every other gate can be built from NAND gates alone.

The answer

The NAND gate

NAND stands for NOT-AND: it is an AND gate followed by a NOT gate. Its output is the inverse of the AND output, so it is logic 0 only when all inputs are 1, and logic 1 in every other case. The Boolean expression is:

Truth table: $00 \to 1$, $01 \to 1$, $10 \to 1$, $11 \to 0$. The symbol is the AND shape with a small circle (bubble) on its output to show the inversion.

The NOR gate

NOR stands for NOT-OR: it is an OR gate followed by a NOT gate. Its output is the inverse of the OR output, so it is logic 1 only when all inputs are 0, and logic 0 in every other case. The Boolean expression is:

Truth table: $00 \to 1$, $01 \to 0$, $10 \to 0$, $11 \to 0$. The symbol is the OR shape with a bubble on its output.

Why NAND is universal

A gate is universal if every other logic function can be built from copies of it alone. The NAND gate is universal: NOT, AND, OR and NOR can all be made using only NAND gates. For example:

• A NOT gate is made by joining both inputs of a NAND together, so it computes the inverse of a single input.
• An AND gate is a NAND followed by a NAND-as-inverter, which cancels the inversion.

Because a manufacturer can build an entire logic system from one type of gate, NAND (and equally NOR) gates simplify design and production. This universality is a key idea of digital electronics.

Making a NOT from a NAND

The most useful trick is turning a NAND into an inverter: tie both inputs together (or hold one input at logic 1). Then the output is the inverse of the common input, exactly the NOT function. This is the building block that lets NAND gates make every other gate.

Examples in context

Example 1. A single-chip logic design. A designer who needs a mix of AND, OR and NOT functions can build the whole circuit from one type of integrated circuit packed with NAND gates. Because NAND is universal, only one part number is stocked, which cuts cost and simplifies the layout. This is universality put to practical use.

Example 2. A NOR-based alarm reset. A NOR gate outputs 1 only when both inputs are 0, which makes it natural for "all clear" logic: the output goes high only when every sensor reads 0. The same gate, used the other way, is a building block for memory latches, showing how the inverted gates earn their keep.

Try this

• Cue. State the only input combination for which a two-input NAND gate outputs 0. When both inputs are 1; for every other combination the NAND output is 1.

• Cue. Write the Boolean expression for a NOR gate with inputs $A$ and $B$. $Q = \overline{A + B}$, the inverse of the OR output.

• Cue. Explain how to make a NOT gate from a single NAND gate. Connect both NAND inputs to the same signal; the output is then the inverse of that input, which is the NOT function.