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verum-nimply logic: xor

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minjaesong
2023-11-22 18:59:41 +09:00
parent 660e492035
commit c514f63f8f
1 changed files with 23 additions and 3 deletions
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work_files/verum-nimply_logic.md
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@@ -34,14 +34,14 @@ This document will use the convention where the Collector is placed on the lefth
### Falsum ### Falsum
⊥ = (⊤ ↛ ⊤) ⊥ = (⊤ ↛ ⊤)
### NOT p
= (⊤ ↛ p)
### Buffer ### Buffer
p = (p ↛ ⊥) p = (p ↛ ⊥)
= NOT (NOT p) = NOT (NOT p)
= (⊤ ↛ (⊤ ↛ p)) = (⊤ ↛ (⊤ ↛ p))
### NOT p
= (⊤ ↛ p)
### p AND q ### p AND q
= (p ↛ (NOT q)) = (p ↛ (NOT q))
= (p ↛ (⊤ ↛ q)) = (p ↛ (⊤ ↛ q))
@@ -65,3 +65,23 @@ Method 2:
= (⊤ ↛ ((⊤ ↛ p) ↛ (⊤ ↛ (⊤ ↛ q)))) = (⊤ ↛ ((⊤ ↛ p) ↛ (⊤ ↛ (⊤ ↛ q))))
= (⊤ ↛ ((⊤ ↛ p) ↛ q)) = (⊤ ↛ ((⊤ ↛ p) ↛ q))
Note: For the most cases, an OR gate can be substituted using merging wires and buffers. Note: For the most cases, an OR gate can be substituted using merging wires and buffers.
### p NOR q
= NOT (p OR q)
= NOT (NOT(p) AND NOT(q))
= NOT(p) AND NOT(q)
= ((⊤ ↛ p) ↛ (⊤ ↛ (⊤ ↛ q)))
= ((⊤ ↛ p) ↛ q)
### p XOR q
= (p OR q) AND NOT(p AND q)
= (p OR q) AND (p NAND q)
= (⊤ ↛ ((⊤ ↛ p) ↛ q)) AND (⊤ ↛ (p ↛ (⊤ ↛ q)))
= ((⊤ ↛ ((⊤ ↛ p) ↛ q)) ↛ (⊤ ↛ (⊤ ↛ (p ↛ (⊤ ↛ q)))))
= ((⊤ ↛ ((⊤ ↛ p) ↛ q)) ↛ (p ↛ (⊤ ↛ q)))