Adders, Multiplexers, and Decoders

Useful Blocks

Rather than design every circuit from raw gates, engineers assemble standard building blocks — combinational modules so common they are used like Lego pieces.

The adder does binary addition. A half adder adds two bits, producing a sum ($A \oplus B$, the XOR) and a carry ($A \cdot B$, the AND). A full adder adds three bits — two operands plus a carry-in — producing a sum and a carry-out, so full adders can be chained: the carry-out of one feeds the carry-in of the next. Chaining $n$ full adders makes a ripple-carry adder that adds two $n$-bit numbers (with the carry rippling along, which also explains why addition takes longer for wider numbers).

A multiplexer (MUX) is a data selector: it has several data inputs, some select lines, and one output, and it routes the chosen input to the output. A MUX with $2^k$ data inputs needs $k$ select lines. MUXes are how a circuit chooses between sources — the hardware equivalent of an if/else.

A decoder does the opposite: it takes a $k$-bit input and activates exactly one of $2^k$ output lines — the one whose number matches the input. Decoders select a specific memory row or device from an address. Its inverse, an encoder, turns a one-active-line input back into a binary number.

Common pitfall: mixing up a multiplexer and a decoder. A MUX has many inputs and one output (it selects one input to pass through); a decoder has one small input and many outputs (it activates exactly one output line). They are near-opposites — one narrows, the other fans out.

A 4-to-1 multiplexer with four data lines converging; the two select bits choose one path that lights in accent through to the single output — beside a decoder fanning one 2-bit input out to activate one of four out.

$$2^k \text{ inputs} \Rightarrow k \text{ select lines}$$