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Processor Architectures and Boolean Algebra: Question 9

Syllabus 15.2

Structured A2 6 marks

A logic circuit implements the Boolean expression Z = A.B + A.NOT B + NOT A.B, where A and B are single-bit inputs.

(a) Simplify Z algebraically as far as possible, showing each step and naming the Boolean law used at each step. [3]

(b) Complete a truth table showing the value of the original expression Z, and the value of your simplified expression from part (a), for all four combinations of A and B. Confirm that the two columns are identical in every row. [3]

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Worked solution

Part (a): Simplifying Z algebraically

Starting expression: Z = A.B + A.NOT B + NOT A.B

Step 1, factor A from the first two terms (distributive law), then apply the complement and identity laws:

A.B + A.NOT B = A.(B + NOT B)     [distributive law]
              = A.1               [complement law: B + NOT B = 1]
              = A                 [identity law: A.1 = A]

So Z = A + NOT A.B.

Step 2, factor the remaining expression using the distributive law in the form X + NOT X.Y = (X + NOT X).(X + Y), with X = A and Y = B:

A + NOT A.B = (A + NOT A).(A + B)  [distributive law]
            = 1.(A + B)             [complement law: A + NOT A = 1]
            = A + B                 [identity law: 1.X = X]

Final simplified answer: Z = A + B. [3 marks: 1 for correctly reducing A.B + A.NOT B to A using the distributive, complement and identity laws, 1 for correctly reducing A + NOT A.B to A + B using the same set of laws, 1 for the final correct simplified expression Z = A + B with laws correctly named throughout]

Part (b): Verifying with a truth table

Evaluating the original expression Z = A.B + A.NOT B + NOT A.B and the simplified expression A + B for all four combinations of A and B:

ABA.BA.NOT BNOT A.BZ (original)A + B (simplified)
0000000
0100111
1001011
1110011

Checking one row explicitly: for A = 1, B = 0, A.B = 0, A.NOT B = 1.1 = 1, NOT A.B = 0.0 = 0, so Z = 0 + 1 + 0 = 1; and A + B = 1 + 0 = 1, the two columns agree.

The Z (original) column reads 0, 1, 1, 1 and the A + B (simplified) column also reads 0, 1, 1, 1, for AB = 00, 01, 10, 11 respectively. The two columns are identical in every row, confirming that Z = A + B is a correct simplification of the original expression. [3 marks: 1 for the correct original Z column (0, 1, 1, 1), 1 for the correct A + B column (0, 1, 1, 1), 1 for explicitly confirming the two columns match in every row]

Final answers

  • (a) Z = A.B + A.NOT B + NOT A.B simplifies to Z = A + B.
  • (b) Both Z and A + B give 0, 1, 1, 1 for AB = 00, 01, 10, 11. The columns match in every row, confirming the simplification is correct.