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'

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)

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