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Logic Analysis and Design Course, 6803213-3 Second DeMorgans’ Law Proof

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(1)

1 Kingdom of Saudi Arabia

Ministry of Education Umm AlQura University

Adam University College, female branch Computer Science Department

Ψ©ΩŠΨ―ΩˆΨΉΨ³Ω„Ψ§ Ψ©ΩŠΨ¨Ψ±ΨΉΩ„Ψ§ Ψ©ΩƒΩ„Ω…Ω…Ω„Ψ§ Ω…ΩŠΩ„ΨΉΨͺΩ„Ψ§ ةرازو Ψ©ΨΉΩ…Ψ§Ψ¬ Ω‰Ψ±Ω‚Ω„Ψ§ Ω…Ψ£ ΨͺΨ§Ψ¨Ω„Ψ§Ψ·Ω„Ψ§ Ψ±Ψ·Ψ΄ ΨŒΩ…ΨΆΨ£Ψ¨ Ψ©ΩŠΨΉΩ…Ψ§Ψ¬Ω„Ψ§ Ψ©ΩŠΩ„ΩƒΩ„Ψ§

ΩŠΩ„Ω„Ψ’Ψ§ Ψ¨Ψ³Ψ§Ψ­Ω„Ψ§ Ω…ΩˆΩ„ΨΉ Ω…Ψ³Ω‚

First Semester of 2017-2018 Academic Year

Logic Analysis and Design Course, 6803213-3 Second DeMorgans’ Law Proof

The Theorem:

𝐴 βˆ— 𝐡

Μ…Μ…Μ…Μ…Μ…Μ…Μ…= 𝐴̅ + 𝐡̅

The Answer:

For any theorm 𝑋 = π‘Œ, if we can show that 𝑋̅ π‘Œ = 0, and that 𝑋̅ + π‘Œ = 1 then by the complement identities, 𝐴̅ 𝐴 = 0 and 𝐴̅ + 𝐴 = 1, 𝑋̅ = π‘ŒΜ….

By the uniqueness of the complement, 𝑋 = π‘Œ.

Thus the proof consists of showing that (𝐴 βˆ— 𝐡) βˆ— (𝐴̅ + 𝐡̅) = 0; and also that (𝐴 βˆ— 𝐡) + (𝐴̅ + 𝐡̅) = 1.

Prove: (𝐴 βˆ— 𝐡) βˆ— (𝐴̅ + 𝐡̅) = 0

(𝐴 βˆ— 𝐡) βˆ— (𝐴̅ + 𝐡̅) = (𝐴 βˆ— 𝐡) βˆ— 𝐴̅ + (𝐴 βˆ— 𝐡) βˆ— 𝐡̅ By Distributive identitiy = (𝐴 βˆ— 𝐴̅) βˆ— 𝐡 + 𝐴 βˆ— (𝐡 βˆ— 𝐡̅) By associativitiy identitiy = 0 βˆ— 𝐡 + 𝐴 βˆ— 0 By complement identity = 0 + 0 By nulity theorem = 0 By identity theorem (𝐴 βˆ— 𝐡) βˆ— (𝐴̅ + 𝐡̅) = 0

Prove: (𝐴 βˆ— 𝐡) + (𝐴̅ + 𝐡̅) = 1

(𝐴 βˆ— 𝐡) + (𝐴̅ + 𝐡̅) = (𝐴 + 𝐴̅ + 𝐡̅) βˆ— (𝐡 + 𝐴̅ + 𝐡̅ ) By Distributive identitiy (𝐴 βˆ— 𝐡) + (𝐴̅ + 𝐡̅) = (𝐴 + 𝐴̅ + 𝐡̅) βˆ— (𝐡 + 𝐡̅ + 𝐴̅ ) By associativitiy identitiy = (1 + 𝐡̅) βˆ— (1 + 𝐴̅) By complement identity = 1 βˆ— 1 By nulity theorem = 1 By identity theorem (𝐴 βˆ— 𝐡) + (𝐴̅ + 𝐡̅) = 1

Since (𝑨 βˆ— 𝑩) βˆ— (𝑨̅ + 𝑩̅) = 𝟎, and (𝑨 βˆ— 𝑩) + (𝑨̅ + 𝑩̅) = 𝟏, 𝑨 βˆ— 𝑩 is complement of 𝑨̅ + 𝑩̅ meaning that 𝑨 βˆ— 𝑩 = (𝑨̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑩̅); Thus 𝑨 βˆ— 𝑩̅̅̅̅̅̅̅ = (𝑨̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑩̅)

.

The involution theorm states that 𝑨̅̅ = 𝑨. Thus by the involution theorm, (𝑨̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑩̅) = 𝑨̅ + 𝑩̅. This proves the above DeMorgan’s Theorem.

(2)

2 References:

[1] http://www2.nau.edu/~sh295/EE110/deMorganproof.html

Good Luck my Great Students πŸ˜‰ T.Mariah Sami Ahmed Khayat

Teacher Assistant @ Adam University College [email protected]

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