Subtractors

WHAT is a subtractor?

There are two kinds:

Half Subtractor (HS) — subtracts 2 bits, ignores incoming borrow.
Full Subtractor (FS) — subtracts 2 bits plus an incoming borrow-in $B_{in}$ .

Half Subtractor — derive from scratch

We compute $A - B$ for single bits. Enumerate all cases (this is Feynman-level honesty — never guess a truth table, build it):

$A$	$B$	$A-B$	$D$ (diff)	$B_{out}$ (borrow)
0	0	0	0	0
0	1	$-1$ → borrow, result $1$	1	1
1	0	1	1	0
1	1	0	0	0

Why row 2? $0-1$ can't be $-1$ in an unsigned column, so we borrow $2$ : $0+2-1 = 1$ gives $D=1$ , and we mark $B_{out}=1$ because we took from above.

Now read the columns as boolean functions:

$D=1$ when exactly one input is 1 → that's XOR.
$B_{out}=1$ only in the row $A=0,B=1$ → that's $\bar{A}\cdot B$ .

Full Subtractor — derive from scratch

Now include a borrow coming in from the previous (lower) column, $B_{in}$ . We compute $A - B - B_{in}$ .

$A$	$B$	$B_{in}$	$A-B-B_{in}$	$D$	$B_{out}$
0	0	0	0	0	0
0	0	1	$-1$	1	1
0	1	0	$-1$	1	1
0	1	1	$-2$	0	1
1	0	0	1	1	0
1	0	1	0	0	0
1	1	0	0	0	0
1	1	1	$-1$	1	1

Why the $D$ column is XOR of three? The difference bit is 1 whenever an odd number of the three inputs are 1 (check: rows with 1,1,1 or a single 1 give $D=1$ ). Odd-parity of three bits is three-input XOR.

$D = A \oplus B \oplus B_{in}$

Deriving $B_{out}$ algebraically (this mirrors the full-adder carry derivation). Group the $B_{out}=1$ rows using the two-stage idea. Do the subtraction as two half-subtracts:

First $A-B$ : produces intermediate diff $d_1=A\oplus B$ and borrow $b_1=\bar A B$ .
Then $d_1 - B_{in}$ : produces final $D=d_1\oplus B_{in}$ and borrow $b_2=\overline{d_1}\,B_{in}$ .
A borrow is needed overall if either subtract borrowed: $B_{out}=b_1+b_2$ .

$B_{out} = \bar A B + \overline{(A\oplus B)}\,B_{in}$

Simplify $\overline{A\oplus B} = AB + \bar A\bar B$ , so: $B_{out} = \bar A B + (AB+\bar A\bar B)B_{in}$

A neater standard SOP form (verify from the truth table) is:

The 80/20 shortcut: subtraction = addition of 2's complement

$A - B = A + \overline{B} + 1 \pmod{2^n}$

WHY it works: 2's complement of $B$ is $2^n - B$ , so $A + (2^n - B) = 2^n + (A-B)$ . The $2^n$ is just the carry-out you discard, leaving $A-B$ .

Worked examples

Common mistakes

Forecast-then-Verify

Recall Predict before peeking

Q: For FS with $A=1,B=0,B_{in}=1$ , what are $D$ and $B_{out}$ ? Forecast: $1-0-1=0$ , no borrow. Verify: $D=1\oplus0\oplus1=0$ ✓, $B_{out}=\bar1\cdot0+\bar1\cdot1+0\cdot1=0$ ✓.

Recall Feynman: explain to a 12-year-old

Imagine you have candies in stacked jars. Top jar has some, bottom jar has some, and you must take the bottom amount away from the top. If the top jar doesn't have enough, you run upstairs and borrow one big bundle (worth 2 candies in binary) from the next jar up — but now that upper jar owes one back, and that "owes one" note is the borrow-out. The difference bit is just whatever candies are left in the current jar. That's all a subtractor does: leftover candies (difference) and the "I owe upstairs" note (borrow).

Connections

Adders — subtractor shares $D=$ Sum $=$ XOR; borrow mirrors carry with an inverted $A$ .
Twos-Complement-Representation — basis of $A-B=A+\bar B+1$ .
Adder-Subtractor-Circuit — one XOR-controlled unit does both.
XOR-Gate — parity behaviour behind the difference bit.
Ripple-Carry-Chains — cascading FS for multi-bit subtraction.

Flashcards

Half subtractor difference expression

D = A \oplus B

Half subtractor borrow expression

B_{out} = \bar A B

Why is half-subtractor borrow

\bar A B

not

AB

You borrow only when top bit

A=0

and bottom bit

B=1

Full subtractor difference expression

D = A \oplus B \oplus B_{in}

Full subtractor borrow expression

B_{out} = \bar A B + \bar A B_{in} + B B_{in}

(equivalently

\bar A B + \overline{(A\oplus B)}B_{in}

)

Difference between adder-Sum and subtractor-Difference logic

None — both are XOR of the inputs.

How does full-subtractor borrow differ from full-adder carry?

Replace

A

with

\bar A

C_{out}=AB+(A\oplus B)C_{in}

B_{out}=\bar A B+\overline{(A\oplus B)}B_{in}

How to compute

A-B

using an adder?

A + \bar B + 1

(invert B, set carry-in to 1).

Why does 2's-complement subtraction work?

\bar B+1 = 2^n-B

, so

A+2^n-B=2^n+(A-B)

; discard the

2^n

carry.

In multi-bit subtraction, what does the final borrow-out mean?

If 1, the result is negative (

A<B

); if 0,

A\ge B

For FS row

A=0,B=1,B_{in}=1

, give

D,B_{out}

D=0

B_{out}=1

(subtracting

0-1-1=-2

What makes

D

a 3-input XOR in the full subtractor?

Difference is 1 exactly when an odd number of the three inputs are 1 (odd parity).

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, subtractor basically adder ka ulta bhai hai. Jaise addition mein hum "carry" upar bhejte hain, waise hi subtraction mein jab upar wala digit chhota ho jaye to hum next column se "borrow" karte hain — school wala ghatana yaad karo, jab $5-3$ mein upar se ek udhaar leke aate the. Yehi udhaar hardware mein borrow bit ban jaata hai. Half subtractor sirf do bits $A-B$ karta hai: difference $D=A\oplus B$ (bilkul adder ke sum jaisa), aur borrow $B_{out}=\bar A B$ — yani borrow tabhi hota hai jab top bit $A=0$ ho aur $B=1$ ho.

Sabse common galti yehi hai ki log borrow ko $AB$ likh dete hain, kyunki adder ka carry $AB$ hota hai aur diff toh same XOR hai. Par bhai, borrow ke liye top bit ka broke hona zaroori hai (yani $A=0$ ), isliye bar $A$ ke upar lagta hai: $\bar A B$ . Row $A=0, B=1$ ko hamesha check karke confirm karo.

Full subtractor mein ek extra $B_{in}$ (pichle column se aaya udhaar) aata hai. Difference ban jaata hai teen-input XOR: $D=A\oplus B\oplus B_{in}$ , aur borrow $B_{out}=\bar A B+\overline{(A\oplus B)}B_{in}$ — full adder ke carry jaisa hi skeleton, bas $A$ pe bar lag gaya.

Asli maza yeh hai: real processors alag subtractor banate hi nahi. Wo 2's complement trick use karte hain — $A-B = A+\bar B+1$ . Matlab $B$ ke saare bits invert karo aur adder ka carry-in $1$ set kar do. Ek control line se ek hi circuit adder bhi aur subtractor bhi ban jaata hai. Yeh 80/20 point hai — isko pakka yaad rakho, exam aur real design dono mein kaam aayega.

WHAT is a subtractor?

WHY borrow (not carry)?

Half Subtractor — derive from scratch

Full Subtractor — derive from scratch

The 80/20 shortcut: subtraction = addition of 2's complement

Worked examples

Common mistakes

Forecast-then-Verify

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Combinational Circuits

Connections