Boolean algebra help with 3.12. and 3.14e. thank you in advance Boolean Algebra

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Boolean algebra help with 3.12. and 3.14e.
thank you in advance

Boolean Algebra (Conti 3.11 Simplify algebraically to a minimum sum of products (five terms): (A + B'+ C+ E(A + B'+ D'+E(B'+ C+D' + E 3.12 2 Prove algebraically that the following cquati on is valid 3.13 Simplify cach of the following expressions: (a) KLMN + K'I'MN + MN (b) KL'M + MN + LM'N (c) (K + L(K +L'+ N(L+ M+ N 8.14 Factor to obtain a product of sums: M'N'+ IMN (fourterms) (four terms) (a) K, L, M, + KM'NA KI.M + .NI, N' (c) KL + K'L'M +'M'N + LM'N (lour terms) + KI'N +K'MN+ LN (lour terms ) WXY + WX'Y + WYZ XYZ' (three terms) . Temli Multiply out to obiain a sum ol, products;

Explanation / Answer

3.12 A'CD'E+A'B'D'+ABCE+ABD=A'B'D'+ABD+BCD'E

A'CD'E(B+B')+A'B'D'+ABCE(D+D')+ABD=A'B'D'+ABD+BCD'E

  A'CD'EB+A'CD'EB'+A'B'D'+ABCED'+ABCED+ABD=A'B'D'+ABD+BCD'E

A'B'D'(1+CE)+A'CD'EB+ABD(1+CE)+ABCED'=A'B'D'+ABD+BCD'E

A'B'D' +ABD +A'CD'EB+ABCED'=A'B'D'+ABD+BCD'E

A'B'D'+ABD+BCED'(A+A')=A'B'D'+ABD+BCD'E

A'B'D'+ABD+BCED'=A'B'D'+ABD+BCD'E

HENCE LHS =RHS

3.14 (e) WXY + WX'Y + WYZ + XYZ'

WY (X + X' + Z) + XYZ'

WY + XYZ'

Y (W + XZ')

Y (W + X) (W + Z')

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