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Equality Circuit

posted Mar 30, 2018, 12:03 AM by MUHAMMAD MUN`IM AHMAD ZABIDI   [ updated Mar 31, 2018, 12:11 AM ]
How to check if two 4-bit numbers are equal?
First, check if every bit in the same position have the same value.

For example, A=B are equal below.
A = 0011
B = 0011

Let A = {a3a2a1a0} and B = {b3b2b1b0}.
Each column is treated separately.
So a3 = b3 = 0, a2 = b2 = 0, a1 = b1 = 1 and a0 = b0 = 1

If we define xi a signal that is true when a bit pair has equal value then xi = (ai = bi) = ai xnor bi = (ai · bi) or (not ai · not bi).
Then A is equal B because x3 · x2 · x1 · x0 = 1 · 1 · 1 · 1 = 1.

Let's look A B in the following case
A = 0001
B = 0011
Because a3 = b3, a2 = b2, a1 b1  and a0 = b0, x3 · x2 · x1 · x0 becomes 1 · 1 · 0 · 1 which gives a false result.

The following circuit does all that in an instant!




Due to de Morgan's theorem, you an also built the circuit the following way.

E = x3 · x2 · x1 · x0 = not (x3 + x2 + x1 + x0)

The NOR is more commonly used to detect all zero conditions.



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