Logic Analysis and Design Course, 6803213-3 First Homework

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1 Kingdom of Saudi Arabia

Ministry of Education Umm AlQura University

Adam University College, female branch Computer Science Department

ةيدوعسلا ةيبرعلا ةكلمملا ميلعتلا ةرازو ةعماج ىرقلا مأ تابلاطلا رطش ،مضأب ةيعماجلا ةيلكلا

يللآا بساحلا مولع مسق

First Semester of 2017-2018 Academic Year

Logic Analysis and Design Course, 6803213-3 First Homework

Solutions

Question One: 1 Mark

*Demonstrate by means of truth tables the validity of the following identity:

(a) DeMorgan’s theorem for three variables: 𝑋𝑌𝑍̅̅̅̅̅̅ = 𝑋̅ + 𝑌̅ + 𝑍̅

The Answer of Question One:

𝐗 𝐘 𝐙 𝐗𝐘𝐙 𝐗̅ 𝐘̅ 𝐙̅ 𝐗𝐘𝐙̅̅̅̅̅̅ 𝐗̅ + 𝐘̅ + 𝐙̅

0 0 0 0 1 1 1 1 1

0 0 1 0 1 1 0 1 1

0 1 0 0 1 0 1 1 1

0 1 1 0 1 0 0 1 1

1 0 0 0 0 1 1 1 1

1 0 1 0 0 1 0 1 1

1 1 0 0 0 0 1 1 1

1 1 1 1 0 0 0 0 0

The last two columns are the same, so the theorm is true.

Question Two: 1 Mark

*Prove the identity of the following Boolean equation, using algebraic manipulation:

a) 𝑋 ̅ 𝑌̅ + 𝑋̅𝑌 + 𝑋𝑌 = 𝑋̅ + 𝑌 The Answer of Question Two:

The proof steps Justification 𝑋 ̅ 𝑌̅ + 𝑋̅𝑌 + 𝑋𝑌 =

𝑋 ̅ (𝑌̅ + 𝑌) + 𝑋𝑌 = Distributive 𝑋 ̅ (1) + 𝑋𝑌 = 𝑋 + 𝑋̅ = 1 𝑋 ̅ + 𝑋𝑌 = 𝑋 ∗ 1 = 𝑋 (𝑋 ̅ + 𝑋)(𝑋 ̅ + 𝑌) = Distrivbutive (1)(𝑋 ̅ + 𝑌) = 𝑋 + 𝑋̅ = 1 𝑋 ̅ + 𝑌 𝑋 ∗ 1 = 𝑋

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2 Question Three: 1 Mark (Optional)

*Simplify the following Boolean expression to expression containing a minimum number of literals:

a) 𝐴̅𝐶̅ + 𝐴̅ 𝐵 𝐶 + 𝐵̅ 𝐶

The Answer of Qestion Three:

𝐴̅𝐶 + 𝐴̅𝐵𝐶 + 𝐵̅𝐶

𝐴̅𝐵𝐶̅ + 𝐴̅𝐵̅𝐶̅ + 𝐴̅𝐵𝐶 + 𝐴𝐵̅𝐶 + 𝐴̅𝐵̅𝐶

𝐴̅𝐵𝐶 + 𝐴̅𝐵𝐶̅ + 𝐴𝐵̅𝐶 + 𝐴̅𝐵̅𝐶̅ + 𝐴̅𝐵̅𝐶 + 𝐴𝐵̅𝐶 𝐴̅ + 𝐵̅𝐶

𝑨𝑩̅̅̅̅ 𝑨̅𝑩 𝑨𝑩 𝑨𝑩̅

𝑪̅ 1 1

𝑪 1 1 1

Good Luck my Great Students 😉 T.Mariah Sami Ahmed Khayat

Teacher Assistant @ Adam University College [email protected]