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Boolean Algebra

In Boolean algebra, only two states of variables (0 and 1) are allowed. The variables (ABC…) of Boolean Algebra are subjected to following three operations:
 
OR operation Represented by (+) sign.
 
109192.png 
Fig. 40
 
Boolean expression, Y = A + B
 
When switch A or B is closed, bulb glows.
 
AND operation Represented by (·) sign.
 
Boolean expression, Y = A · B
 
When switches A and B both are closed, bulb glows.
 
109209.png 
Fig. 41
 
NOT operation Represented by bar over the variables
 
Boolean expression, 107242.png 
 
109221.png
Fig. 42

Basic Boolean postulates and laws

  • Boolean postulates: 0 + A = A, 1 · A = A,
     
    1 + A = 1, 0 · A = 0,
     
    107248.png
  • —Identity law: A + A = A, A · A = A
  • —Negation law: 107254.png
  • —Commutative law: A + B = B + A, A · B = B · A
  • —Associative law: (A+B) + C = A + (B+C),
     
    (A · B) · C = A · (B · C)
  • —Distributive law: A · (B+C) = A · B + A · C
     
    (A + B) · (A + C) = A + BC
  • —Absorption laws: A + A·B = A, A · (A + B) = A
     
    107260.png
  • Boolean identities: 107266.png, 107272.png
     
    107281.png,
     
    107289.png
De Morgan’s theorem It states that the complement of the whole sum is equal to the product of individual complements and vice versa, i.e., 107298.png and 107304.png.




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