Homework 1 Prove the following theorems algebraically: a) X(X\' + Y) = XY b) X +

Question

Homework 1

Prove the following theorems algebraically:

a) X(X' + Y) = XY

b) X + XY = X

c) XY + XY' = X

d) (A + B)(A + B') = A

Problem 2

Multiply out and simplify to obtain a sum of products:

a) (A + B)(C + B)(D' + B)(ACD' + E)

b) (A' + B + C')(A' + C' + D)(B' + D')

Problem 3

Factor each of the following expressions to obtain a product of sums:

a) AB + C'D'

b) XYZ + W'Z + XQ'Z

Problem 4

Simplify each of the following expressions by applying one of the theorems. State the theorem used:

a) (A' + B' + C)(A' + B' + C)'

b) (A'BF + CD')(A'BF + CEG)

c) A' (B + C)(D'E + F)' + (D'E + F)

Problem 5

Prove the following equations using truth tables:

a) (X + Y)(X' + Z) = XZ + X'Y

b) (X + Y)(Y + Z)(X' + Z) = (X + Y)(X' + Z)

c) W'XY + WZ = (W' + Z)(W + XY)

Explanation / Answer

2. Problem 2

a) (A + B)(C + B)(D' + B)(ACD' + E)

(AC+AB+BC+B)(ACD'+D'E+ABCD'+BE)

=(AC+(A+C+1)B)(ACD'+D'E+ABCD'+BE)

=(AC+B)(ACD'+D'E+ABCD'+BE)

=ACD'+ACD'E+ABCD'+ACBE+ABCD'+BD'E+ABCD'+BE

=ACD'(1+E+B+B+B)+BE(1+D'+AC)

=ACD'+BE

b) (A' + B + C')(A' + C' + D)(B' + D')

(A'+B+C')(A'B'+A'D'+B'C'+C'D'+DB'+DD')

=(A'+B+C')(A'B'+A'D'+B'C'+C'D'+DB')

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