How is the final output X formed from the four bit-equality signals?

AND of all four, so X = 1 only if every pair matches

OR of the four, so X = 1 if any pair matches

AND of only the two MSB signals

Why does the answer use a bit-wise method instead of a K-map?

The truth table has 256 rows with only 16 having X = 1, which makes a K-map impractical

Asked 2/22
2082 Baisakh · 5 marks

4-bit equality comparator

Short answer

Compare each bit pair with an XNOR gate: xi = Ai ⊙ Bi. AND all four outputs: X = x3·x2·x1·x0. X=1 iff A=B. Alternatively, XOR each pair and NOR the results: X = ̅(A3⊕B3)+...+(A0⊕B0).

Design a circuit that compares two 4-bit numbers, A and B, to check if they are equal. The circuit has one output X, so that X = 1 if A = B and X = 0 if A ≠ B.

Two 4-bit numbers $A = A_3A_2A_1A_0$ and $B = B_3B_2B_1B_0$ are equal only when every pair of corresponding bits is equal. Equality of one pair is detected by an XNOR gate.

Bit-equality signals

x_i = A_i \odot B_i = A_iB_i + A_i'B_i', \quad i = 0, 1, 2, 3

$A_i$	$B_i$	$x_i$
0	0	1
0	1	0
1	0	0
1	1	1

Output

$X = 1$ only if all four $x_i = 1$ :

X = x_3x_2x_1x_0 = (A_3 \odot B_3)(A_2 \odot B_2)(A_1 \odot B_1)(A_0 \odot B_0)

Equivalent form with XOR and NOR:

X = \overline{(A_3 \oplus B_3) + (A_2 \oplus B_2) + (A_1 \oplus B_1) + (A_0 \oplus B_0)}

A full truth table would have $2^8 = 256$ rows with $X = 1$ in only 16 of them ( $A = B$ ), so the bit-wise method is used instead of a K-map.

Logic circuit

 A3 --+
 B3 --+--[XNOR]-- x3 --+
 A2 --+                |
 B2 --+--[XNOR]-- x2 --+
                       +--[AND 4]--- X
 A1 --+                |
 B1 --+--[XNOR]-- x1 --+
 A0 --+                |
 B0 --+--[XNOR]-- x0 --+

Gates: four 2-input XNOR gates and one 4-input AND gate.

Examples

A	B	$x_3x_2x_1x_0$	X
1011	1011	1111	1
1011	1001	1101	0
0110	1110	0111	0

This is the "A = B" part of the 7485 4-bit magnitude comparator.

✓ Answer checked by Claude Opus 5.5 (high effort)

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4-bit equality comparator

Bit-equality signals

Output

Logic circuit

Examples

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