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21 July, 15:54

Simplify the following Boolean expressions using algebraic manipulation.

a. F (a, b) = a'b' + ab + ab'

b. F (r, s, t) = r' + rt + rs' + rs't'

c. F (x, y) = (x + y) ' (x' + y') '

d. F (a, b, c) = b' + bc + ab + a'bc'

e. F (a, b, c, d) = ac + b'd + a'bc + b'cd'

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Answers (2)
  1. 21 July, 16:20
    0
    a) De Morgan's law: i) a * b = ' (a' + b)

    ii) a + b = ' ('a + 'b)

    b) Associativity: (r * s') + (s * t) + (t' * r)

    c) Commutativity: i) x + y = y + x

    ii) x * y = y * x

    d) Distributivity: a * (b + c) = (a * b) + (a + c)
  2. 21 July, 17:45
    0
    Simplification of the expressions:

    a) a'b' + ab + ab' = a + b'

    b) r' + rt + rs' + rs't' = r' + s' + t

    c) (x + y) ' (x' + y') ' = False

    d) b' + bc + ab + a'bc' = True

    e) ac + b'd + a'bc + b'cd' = c + b'd

    Explanation:

    The step by step solution for each expression will use the following laws of Boolean Algebra:

    Idempotent Law:

    aa=a

    a+a=a

    Associative Law

    :

    (ab) c=a (bc)

    (a+b) + c=a + (b+c)

    Commutative Law

    :

    ab=ba

    a+b=b+a

    Distributive Law

    :

    a (b+c) = ab+ac

    a + (bc) = (a+b) (a+c)

    Identity Law

    :

    a*0=0 a*1=a

    a+1=1 a+0=a

    Complement Law

    :

    aa'=0

    a+a'=1

    Involution Law

    :

    (a') '=a

    DeMorgan's Law

    :

    (ab) '=a'+b'

    (a+b) '=a'b'

    Absorption Law:

    a + (ab) = a

    a (a+b) = a

    (ab) + (ab') = a

    (a+b) (a+b') = a

    a + (a'b) = a+b

    a (a'+b) a*b

    Step by step Solution:

    a) F (a, b) = a'b' + ab + ab'

    a (b+b') + a'b' Commutative Law

    a+a'b Complement Law

    F (a, b) = a+b' Absorption Law

    b) F (r, s, t) = r' + rt + rs' + rs't'

    (r'+rs') + rt+rs't' Absorption Law

    r'+s'+rt+rs't' Distributive Law

    r'+s'+rt+s' Absorption Law

    r'+s'+rt Absorption Law

    F (r, s, t) = r'+s'+t Absorption Law

    c) F (x, y) = (x + y) ' (x' + y') '

    (x'y') (x''y'') DeMorgan's Law

    (x'y') xy Involution Law

    x' (y'x) y Associative Law

    x' (xy') y Commutative Law

    (x'x) (y'y) Associative Law

    (0) (0) Complement Law

    F (x, y) = False

    d) F (a, b, c) = b' + bc + ab + a'bc'

    b'+c+b (a+a'c') Absorption Law

    b'+c+b (a+c') Absorption Law

    b'+c+ba+bc' Distributive Law

    (b'+ba) + (c+bc') Associative Law

    b'+a+c+b Absorption Law

    1+a+c Complement Law

    F (a, b, c) = True

    e) F (a, b, c, d) = ac + b'd + a'bc + b'cd'

    ac+a'bc+b'd+b'cd' Commutative Law

    c (a+a'b) + b' (d+cd') Associative and Distributive Law

    c (a+b) + b' (d+c) Absorption Law

    ac+bc+b'd+b'c Distributive Law

    ac + (bc+b'c) + b'd Associative and Commutative Law

    ac+c (b+b') + b'd Associative and Distributive Law

    ac+c*1+b'd Complement Law

    c (a+1) + b'd Distributive and Identity Law

    F (a, b, c, d) = c+b'd
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