Number Systems and Boolean Algebra

Absorption and the second distributive law collapse most expressions in one step. The distributive form $x+yz=(x+y)(x+z)$ has no counterpart in ordinary algebra, which is why it is easy to miss.

A prime implicant is a group that cannot be enlarged. It is essential when some minterm is covered by it alone — find those minterms first and the rest of the cover follows.

With the select lines driven by the variables, the data inputs of a MUX are literally the truth-table column of the function you want.

The pull-up network is the dual of the pull-down: series in one becomes parallel in the other. Static CMOS draws no steady current in either state — power is dissipated only while switching.

Two's complement is used because addition and subtraction become the same operation and there is only one zero. Its range is asymmetric — one more negative value than positive — and that asymmetry is what most questions turn on.

A ripple-carry adder is slow for the same reason it is simple: bit $n$ cannot settle until bit $n-1$ has. Carry look-ahead breaks the chain by computing every carry directly from generate and propagate terms.

A decoder generates every minterm of its inputs, so a decoder plus an OR gate implements any function directly from its truth table. An encoder does the reverse and needs a priority scheme, because two simultaneous inputs would otherwise produce a meaningless code.