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Introduction to Boolean Algebra: Definitions, Interpretation, and Theorems

This lecture provides an introduction to Boolean algebra, explaining definitions, interpretation in set and logic operations, and theorems. It also covers multi-valued Boolean algebra and expression transformations.

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Introduction to Boolean Algebra: Definitions, Interpretation, and Theorems

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  1. CSE 20: Lecture 7Boolean AlgebraCK Cheng4/21/2011

  2. Outline • Introduction • Definitions • Interpretation in Set Operations • Interpretation in Logic Operations • Theorems and Proofs • Multi-valued Boolean Algebra • Expression Transformations

  3. Introduction Boolean algebra is used in computers for arithmetic & logic operations. Eg: 1. if a is true, then y = b, else y = c. 2. y is true if a and b are true. 3. y is true if a or b is true.

  4. Introduction We use binary bits to represent true or false. A=1: A is true A=0: A is false We use AND, OR, NOT gates to operate the logic. NOT gate inverts the value (flip 0 and 1) y = NOT (A)= A’ A A’

  5. Introduction OR gate: Output is true if either input is true y= A OR B A A OR B B

  6. Introduction AND gate: Output is true only if all inputs are true y= A AND B A A AND B B

  7. Introduction A Half Adder: Carry = A AND B Sum = (A AND B’) OR (A’ AND B) Carry: Carry = A AND B A B

  8. Introduction A Half Adder: Carry = A AND B Sum = (A AND B’) OR (A’ AND B) Sum: A’ A A’ and B (A and B’) or (A’ and B) B A and B’ B’

  9. Definition Boolean Algebra: A set of elements B with two operations. + (OR, U, ˅ ) * (AND, ∩, ˄ ), satisfying the following 4 laws. P1. Commutative Laws: a+b = b+a; a*b = b*a, P2. Distributive Laws: a+(b*c) = (a+b)*(a+c); a*(b+c)= (a*b)+(a*c), P3. Identity Elements: Set B has two distinct elements denoted as 0 and 1, such that a+0 = a; a*1 = a, P4. Complement Laws: a+a’ = 1; a*a’ = 0.

  10. Interpretation in Set Operations • Set: Collection of Objects • Example: • A = {1, 3, 5, 7, 9} • N = {x | x is a positive integer}, e.g. {1, 2, 3,…} • Z = {x | x is an integer}, e.g. {-1, 0, 4} • Q = {x | x is a rational number}, e.g. {-0.75, ⅔, 100} • R = {x | x is a real number}, e.g. {π, 12, -⅓} • C = {x | x is a complex number}, e.g. {2 + 7i} • Ф = {} or empty set

  11. P1. Commutative Laws in Venn Diagram A+B A*B B A A B

  12. P2. Distributive Laws • A* (B+C) = (A*B) + (A*C) C A B

  13. P2. Distributive Laws • A+ (B*C) = (A+B) * (A+C) C A B

  14. P3. Identity Elements • 0 = {} • 1 = Universe of the set • A+0 = A • A*1 = A

  15. P4: Complement • A+A’= 1 • A*A’= 0 A A’

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