mux
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purpose of an mux is to combine inputs using a control. Accomplishes ...
select the data inputs which will be processed and stored.

Study the inputs to the ..... and how outputs are determined. For example,

z = a'Isub(0) + a*Isub(1)

As we did in recitation lab <url::> the outputs of an adder can be combined
with an MUX to yield a 4-bit bus.

decoder

A decoder is the opposite of a multiplexer.

It decodes multiple lines from a line which was created by a multiplexer.
(Remember the telephone lines where the T1 can hold 24 simultaneous
conversations, these phone conversations were put on the T1 by a multiplexer,
and decoded by a decoder).

The struture of a decoder is multiple NAND gates, there will be 2(ⁿ⁾ outputs
Some of those may be don't cares, however. The inputs also are split by an
inverters. The NAND gates ensures that only one of the outputs will be a zero
so the rest are ones, hence a unique output is generated (actually 2ⁿ⁾
outputs are generated.