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education/computer engineering/ECE2700/Binary Logic.md
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# Properties of Boolean Algebra
| 17. Consensus | | | | |
| Number | Col. A | Col. A Description | Col. B | Col. B Description |
| ---------------------- | --------------------------------------------------------------------------------- | ------------------ | ----------------------------------------------------------------------------------- | ------------------ |
| 1. | $0 \cdot 0 = 0$ | | $1 + 1 = 1$ | |
| 2. | $1 \cdot 1 = 1$ | | $0 + 0 = 0$ | |
| 3. | $0 \cdot 1 = 1 \cdot 0 = 0$ | | $1 + 0 = 0 + 1 = 1$ | |
| 4. | if $x = 0$ then $\overline{x} = 1$ | | if $x = 1$ then $\overline{x} = 0$ | |
| 5. | $x \cdot 0 = 0$ | | $x + 1 = 1$ | |
| 6. | $x \cdot 1 = x$ | | $x + 0 = x$ | |
| 7. | $x \cdot x = x$ | | $x + x = x$ | |
| 8. | $x \cdot \overline{x} = 0$ | | $$x + \overline{x} = 1$ | |
| 9. | $\overline{\overline{x}} = x$ | | | |
| 10. Commutative | $x \cdot y = y \cdot x$ | | $x + y = y + x$ | |
| 11. Associative | $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ | | $x + (y + z) = (x + y) +z$ | |
| 12. Distributive | $x \cdot (y +z) = x \cdot y + x \cdot z$ | | $x + y \cdot z = (x + y) \cdot (x + z$ | |
| 13. Absorption | $x + x \cdot y = x$ | | $x \cdot (x + y) = x$ | |
| 14. Combining | $x \cdot y + x \cdot \overline{y} = x$ | | $(x + y) \cdot (x + \overline{y}) = x$ | |
| 15. DeMorgan's Theorem | $\overline{x \cdot y} = \overline{x} + \overline{y}$ | | $x + y = \overline{x} \cdot \overline{y}$ | |
| 16. | $x + \overline{x} \cdot y = x + y$ | | $x \cdot (\overline{x} + y) = x \cdot y$ | |
| 17. Consensus | $x \cdot y + y \cdot z + \overline{x} \cdot z = x \cdot y + \overline{x} \cdot z$ | | $(x + y) \cdot (y + z) \cdot (\overline{x} + z) = (x + y) \cdot (\overline{x} + z)$ | |
# Logic Gates