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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 = 0011B = 0011Let A = {a3a2a1a0} and B = {b3b2b1b0}.Each column is treated separately.So a3 = b3 = 0, a2 = b2 = 0, a1 = b1 = 1 and a0 = b0 = 1If 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 caseA = 0001B = 0011Because 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. 