Question

Convert these expressions to both sum of products and product of sums, minimizing where possible

1. abc+(d+e)c’+abc’

2. a(b+c(d+e))+d

Answer 1

Question 1:

abc + (d + e)c’ + abc’

ab(c + c’) + dc’ + ec’

Now, c + c’ = 1

On simplifying we get,

ab + dc’ + ec’

Since,x + x’ = 1, there will be no change in multiplying with it.

ab(c+c’)(d+d’)(e+e’) + (a+a’)(b+b’)c’d(e+e’) + (a+a’)(b+b’)c’(d+d’)e

Let us divide the expression into three parts,

P = ab(c+c’)(d+d’)(e+e’)

Q = (a+a’)(b+b’)c’d(e+e’)

R = (a+a’)(b+b’)c’(d+d’)e

P = abcde + abcde’ + abcd’e + abcd’e’ + abc’de + abc’de’ + abc’d’e + abc’d’e’

P = ?( 31, 30, 29, 28, 27, 26, 25, 24)

Q = (ab + ab’ + a’b + a’b’)c’d(e+e’)

Q = (abc’d + ab’c’d + a’bc’d + a’b’c’d)(e + e’)

Q = abc’de + ab’c’de + a’bc’de + a’b’c’de + abc’de’ + ab’c’de’ + a’bc’de’ + a’b’c’de’

Q = ?( 27, 19, 11, 3, 26, 18, 10, 2)

R = (ab + ab’ + a’b + a’b’)c’(d+d’)e

R = (abc’ + ab’c’ + a’bc’ + a’b’c’)(d + d’)e

R = (abc’d + ab’c’d + a’bc’d + a’b’c’d + abc’d’ + ab’c’d’ + a’bc’d’ + a’b’c’d’)e

R = abc’de + ab’c’de + a’bc’de + a’b’c’de + abc’d’e + ab’c’d’e + a’bc’d’e + a’b’c’d’e

R = ?( 19, 11, 3, 25, 17, 9, 1)

Now on adding P, Q and R we get,

?( 1, 2, 3, 9, 10, 11, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31)

It can also be written as,

?( 4,5,6,7,8,12,13,14,15,16,20,21,22,23)

Question 2:

Expression = ab + acd + ace + d

Let P = ab, Q = acd, R = ace , S = d

P = ab(c+c’)(d+d’)(e+e’)

P = abcde + abcde’ + abcd’e + abcd’e’ + abc’de + abc’de’ + abc’d’e + abc’d’e’

P = ?( 31, 30, 29, 28, 27, 26, 25, 24)

Q = acd(b+b’)(e+e’)

Q = acd(be + be’ + b’e + b’e’)

Q = abcde + abcde’ + ab’cde + ab’cde’

Q = ?(31,30,23,22)

R = a(b+b’)c(d+d’)e

R = ace(bd + bd’ + b’d + b’d’)

R = abcde + abcd’e + ab’cde + ab’cd’e

R = ?(31,29,23,21)

S = (a+a’)(b+b’)(c+c’)d(e+e’)

S = abcde + abcde’ + abc’de + abc’de’ + ab’cde + ab’cde’ + ab’c’de + ab’c’de’ + a’bcde + a’bcde’ + a’bc’de + a’bc’de’ + a’b’cde + a’b’cde’ + a’b’c’de + a’b’c’de’

S = ?(31,30,27,26,23,22,19,18,15,14,11,10,7,6,3,2)

On adding P,Q,R and S, we get

?(2,3,6,7,10,11,14,15,18,19,21,22,23,26,27,29,30,31)

This can also be written as,

?(1,4,5,8,9,12,13,16,17,20,24,25,28)

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