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'

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

Explanation:

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)