Adders, Multiplexers, and Decoders

Building Blocks: Adders

Instead of raw gates, engineers use standard building blocks: reliable combinational modules acting like Lego pieces.

The half adder adds two bits to output a sum ($A \oplus B$) and a carry ($A \cdot B$). A full adder adds three bits including a carry-in, outputting a sum and carry-out.

Chaining $n$ full adders builds a ripple-carry adder for $n$-bit numbers. Wider numbers take longer because the carry must ripple through every stage.

MUX vs. Decoder

A multiplexer (MUX) is a data selector routing one of $2^k$ inputs to a single output using $k$ select lines ($2^k \text{ inputs} \Rightarrow k \text{ select lines}$). It acts as hardware if/else.

A decoder does the opposite: a $k$-bit input activates exactly one of $2^k$ outputs. Its inverse is an encoder.

Common pitfall: MUXes narrow many inputs to one; decoders fan a small input out to many outputs.