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Digital Fundamentals

Digital Fundamentals. CHAPTER 4 Boolean Algebra and Logic Simplification. Boolean Operations and Expressions . Addition 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1. Multiplication 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1. Boolean Operations and Expressions.

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Digital Fundamentals

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  1. Digital Fundamentals CHAPTER 4 Boolean Algebra and Logic Simplification

  2. Boolean Operations and Expressions

  3. Addition 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 Multiplication 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1 Boolean Operations and Expressions

  4. Laws and Rules of Boolean Algebra

  5. Laws Boolean Algebra • Commutative Laws • Associative Laws • Distributive Law

  6. Laws of Boolean Algebra • Commutative Law of Addition: A + B = B + A

  7. Laws of Boolean Algebra • Commutative Law of Multiplication: A * B = B * A

  8. Laws of Boolean Algebra • Associative Law of Addition: A + (B + C) = (A + B) + C

  9. Laws of Boolean Algebra • Associative Law of Multiplication: A * (B * C) = (A * B) * C

  10. Laws of Boolean Algebra • Distributive Law: A(B + C) = AB + AC

  11. Rules of Boolean Algebra

  12. OR Truth Table Rules of Boolean Algebra • Rule 1

  13. OR Truth Table Rules of Boolean Algebra • Rule 2

  14. AND Truth Table Rules of Boolean Algebra • Rule 3

  15. AND Truth Table Rules of Boolean Algebra • Rule 4

  16. OR Truth Table Rules of Boolean Algebra • Rule 5

  17. OR Truth Table Rules of Boolean Algebra • Rule 6

  18. AND Truth Table Rules of Boolean Algebra • Rule 7

  19. AND Truth Table Rules of Boolean Algebra • Rule 8

  20. Rules of Boolean Algebra • Rule 9

  21. OR Truth Table AND Truth Table Rules of Boolean Algebra • Rule 10: A + AB = A

  22. OR Truth Table AND Truth Table Rules of Boolean Algebra • Rule 11:

  23. OR Truth Table AND Truth Table Rules of Boolean Algebra • Rule 12: (A + B)(A + C) = A + BC

  24. DeMorgan’s Theorem

  25. Theorem 1 Theorem 2 DeMorgan’s Theorems • Remember: • “Break the bar, change the sign”

  26. Standard Forms of Boolean Expressions

  27. Standard Forms of Boolean Expressions • The sum-of-product (SOP) form Example: X = AB + CD + EF • The product of sum (POS) form Example: X = (A + B)(C + D)(E + F)

  28. The Karnaugh Map

  29. 3-Variable Example 3-Variable Karnaugh Map The Karnaugh Map

  30. 4-Variable Karnaugh Map 4-Variable Example The Karnaugh Map

  31. 5-Variable Karnaugh Mapping The Karnaugh Map

  32. VHDL

  33. VHDL Operators and or not nand nor xor xnor VHDL Elements entity architecture VHDL

  34. Entity Structure Example: entity AND_Gate1 is port(A,B:in bit:X:out bit); end entity AND_Gate1 VHDL

  35. Architecture Example: architecture LogicFunction of AND_Gate1 is begin X<=A and B; end architecture LogicFunction VHDL

  36. Boolean Expressions in VHDL AND X <= A and B; OR X <= A or B; NOT X <= A not B; NAND X <= A nand B; NOR X <= A nor B; XOR X <= A xor B; XNOR X <= A xnor B; Hardware Description Languages (HDL)

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