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Jeff Shelton – 16 January 2015
Jeff Shelton – 16 January 2015 2
Jeff Shelton – 16 January 2015
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Jeff Shelton – 16 January 2015
Jeff Shelton – 16 January 2015 5
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Jeff Shelton – 16 January 2015
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Jeff Shelton – 16 January 2015
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Jeff Shelton – 16 January 2015
510 = 1012
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Jeff Shelton – 16 January 2015
Jeff Shelton – 16 January 2015
𝑥 or 𝑥′ /𝑥
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𝒙 𝒙
0 1
1 0
𝒙
0 1
𝒙 1 0
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Jeff Shelton – 16 January 2015
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𝒙 𝒚 𝒙 • 𝒚
0 0 0
0 1 0
1 0 0
1 1 1
𝒙 AND 𝒚 𝒙
0 1
𝒚 0 0 0
1 0 1
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Jeff Shelton – 16 January 2015
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𝒙 𝒚 𝒙 + 𝒚
0 0 0
0 1 1
1 0 1
1 1 1
𝒙 OR 𝒚 𝒙
0 1
𝒚 0 0 1
1 1 1
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Jeff Shelton – 16 January 2015
⊕
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𝒙 𝒚 𝒙 ⊕ 𝒚
0 0 0
0 1 1
1 0 1
1 1 0
𝒙 XOR 𝒚 𝒙
0 1
𝒚 0 0 1
1 1 0
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Jeff Shelton – 16 January 2015
𝑥 + 𝑦 = 𝑦 + 𝑥 𝑥 ⋅ 𝑦 = 𝑦 ⋅ 𝑥
𝑥 + 𝑦 + 𝑧 = 𝑥 + 𝑦 + 𝑧 𝑥 ⋅ 𝑦 ⋅ 𝑧 = 𝑥 ⋅ 𝑦 ⋅ 𝑧
𝑥 ⋅ 𝑦 + 𝑧 = 𝑥 ⋅ 𝑦 + 𝑥 ⋅ 𝑧 𝑥 + 𝑦 ⋅ 𝑧 = 𝑥 + 𝑦 ⋅ 𝑥 + 𝑧
𝑥 + 0 = 𝑥 𝑥 ⋅ 1 = 𝑥
𝑥 ⋅ 𝑥 = 0 𝑥 + 𝑥 = 1
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(Axioms define domain characteristics, from which other ‘truths’ can be derived.)
Jeff Shelton – 16 January 2015
𝑥 + 𝑥 = 𝑥 𝑥 ⋅ 𝑥 = 𝑥
𝑥 + 𝑥 ⋅ 𝑦 = 𝑥 𝑥 ⋅ (𝑥 + 𝑦) = 𝑥
𝑥 + 𝑥 ⋅ 𝑦 = 𝑥 + 𝑦
𝑥 ⋅ 𝑥 + 𝑦 = 𝑥 ⋅ 𝑦
(𝑥 + 𝑦) = 𝑥 ⋅ 𝑦
(𝑥 ⋅ 𝑦) = 𝑥 + 𝑦
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(Theorems are proven through algebraic manipulation or exhaustive substitution.)
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Jeff Shelton – 16 January 2015
0 0
0
1
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Jeff Shelton – 16 January 2015
𝑦 = 𝑪1 + 𝑪2 +⋯+ 𝑪𝑛−1 + 𝑪𝑛
𝑪𝑖 𝑛
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0 0
0
1
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Jeff Shelton – 16 January 2015
𝑦 = 𝐵1 ⋅ 𝐵2 + 𝐵1 ⋅ 𝐵2 + 𝐵1 ⋅ 𝐵2 + 𝐵1 ⋅ 𝐵2
𝑦 = 𝐵1 ⋅ 𝐵2 + 𝐵1 ⋅ 𝐵2 + 𝐵1 ⋅ 𝐵2 + 𝐵1 ⋅ 𝐵2
𝑦 = 𝐵1 ⋅ 𝐵2
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0 0
0
1
Jeff Shelton – 16 January 2015
𝑦 = 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2
𝑦 = 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2
𝑦 = 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2 ⋅ 𝐵1 + 𝐵2
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0 0
0
1
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