Electronics A Guide for Physicists (DRAFT -- in development)

The analysis of digital circuits and their behaviors can be conducted in two ways: 1) the method of exhaustion, and 2) Boolean algebra. The method of exhaustion is a brute-force method in which each gate in a circuit is evaluated in turn based on the inputs values it receives. We used this method in the previous section as we were evaluating circuits constructed solely with NAND gates. Boolean algebra is a mathematical formalism used for two-state variables and the operations that act on these variables. This formalism allows us to analyze circuits containing multiple gates through equation manipulation.

Boolean algebra, like standard algebra, is governed by rules that determine how mathematical operations interact. The following Boolean identities provide these rules by which we must adhere.

1. \(\displaystyle A+B = B+A\)
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2. \(\displaystyle A \cdot B = B \cdot A\)
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3. \(\displaystyle A+(B+C) = (A+B) + C\)
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4. \(\displaystyle A\cdot(B\cdot C) = (A\cdot B)\cdot C \)
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5. \(\displaystyle A\cdot(B+C) = A\cdot B + A\cdot C \)
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6. \(\displaystyle (A+B)\cdot(C+D) = A\cdot C + A\cdot D + B\cdot C + B\cdot D \)
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7. \(\displaystyle \overline{1}=0 \)
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8. \(\displaystyle \overline{0}=1 \)
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9. \(\displaystyle A\cdot 0 = 0 \)
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10. \(\displaystyle A\cdot 1 = A \)
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11. \(\displaystyle A + 0 = A \)
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12. \(\displaystyle A + 1 = 1 \)
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13. \(\displaystyle A + A = A \)
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14. \(\displaystyle A \cdot A = A \)
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15. \(\displaystyle \overline{\overline{A}} = A \)
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16. \(\displaystyle A + \overline{A} = 1 \)
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17. \(\displaystyle A \cdot \overline{A} = 0 \)
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18. \(\displaystyle \overline{A + B} = \overline{A} \cdot \overline{B} \)
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19. \(\displaystyle \overline{A\cdot B} = \overline{A} + \overline{B} \)
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20. \(\displaystyle A + \overline{A}\cdot B = A+B \)
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21. \(\displaystyle \overline{A}+A\cdot B = \overline{A} + B \)
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22. \(\displaystyle A \oplus B = \overline{A}\cdot B + A\cdot\overline{B} = (A+B)\cdot(\overline{A\cdot B}) \)
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23. \(\displaystyle \overline{A\oplus B} = A\cdot B + \overline{A}\cdot\overline{B} \)
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