Boolean Algebra · step 12 of 13

Half adders, full adders & multiplexers

A half adder adds two bits: sum = x ⊕ y and carry = xy. A full adder adds three (including a carry-in), giving sum = x ⊕ y ⊕ cᵢₙ and carry-out = xy + cᵢₙ(x ⊕ y). Chaining n full adders gives a ripple-carry adder. A 2-to-1 multiplexer selects between inputs: out = s′a + sb, and it is a universal building block in its own right.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Start from the truth tableWrite all input combinations and both outputs. The algebra follows from the 1-rows.
Recognise XOR for the sum bitA sum output is 1 when an odd number of inputs are 1 — exactly the XOR of them.
Read the carry as a majorityFor a full adder the carry-out is 1 when at least two of the three inputs are 1.
Chain and analyse delayRipple-carry chains n full adders, so the worst-case delay grows linearly with n — the motivation for carry-lookahead.

Worked example

Derive the sum and carry-out expressions for a full adder from its truth table.

Sum is 1 for input patterns with an odd number of 1s: 001, 010, 100, 111.
That is exactly x ⊕ y ⊕ cᵢₙ.
Carry-out is 1 when at least two inputs are 1: 011, 101, 110, 111.
As a sum of products: xy + xcᵢₙ + ycᵢₙ, which factors to xy + cᵢₙ(x ⊕ y).

Answer. sum = x ⊕ y ⊕ cᵢₙ and cₒᵤₜ = xy + xcᵢₙ + ycᵢₙ — the majority function on three inputs.

Where marks get dropped

These are the specific errors that cost credit on half adders, full adders & multiplexers questions — QED's rubric penalises each of them separately.

Using OR instead of XOR for the sum. OR gives 1 for 1 + 1, which is wrong: the sum bit is 0 with a carry.
Forgetting the carry-in when chaining. The least significant stage may use a half adder, but every later stage needs a full adder.
Ignoring propagation delay. Ripple-carry is correct but slow, and exam questions often ask for the worst-case gate delay.

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Half adders, full adders & multiplexers — frequently asked questions

Why is the carry-out the majority function?

Because adding three bits produces a carry exactly when the total is 2 or 3, which happens iff at least two inputs are 1.

How do multiplexers implement arbitrary functions?

A 2ⁿ-to-1 multiplexer with the variables as select lines can realise any n-variable function by wiring the truth-table outputs to the data inputs — this is Shannon expansion in hardware.

What is carry-lookahead?

A scheme computing generate (gᵢ = xᵢyᵢ) and propagate (pᵢ = xᵢ ⊕ yᵢ) signals so all carries are produced in parallel, reducing delay from O(n) to O(log n).

The rest of Boolean Algebra

Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.

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