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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

用中文

  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0

Software

See: Logic

Truth functions

Inputs | Outputs.

No inputs yield {$2^{(2^0)}$} constant functions as follows:

  • | T Always true
  • | F Always false

This grounds truth and falsehood.

One input A with value T or F yields {$2^{(2^1)}$} truth functions with outputs as follows:

  • TT | T
  • FF | F
  • TF | A
  • FT | Not-A

This stage grounds affirmation and negation.

A, B, each with value T or F, generate {$2^2$} input pairs TT-TF-FT-FF and yield {$2^{(2^2)}$} truth functions with outputs as follows:

  • TTTT | T always true
  • FFFF | F always true
  • TTTF | A-or-B = Not(Not-A-and-Not-B) = if-not-A-then-B = if-not-B-then-A
  • TTFT | A-or-Not-B = Not(Not-A-and-B) = if-B-then-A = if-not-A-then-not-B
  • TFTT | Not-A-or-B = Not(A-and-Not-B) = if-A-then-B = if-not-B-then-not-A
  • FTTT | Not-A-or-Not-B = Not(A-and-B) = if-A-then-not-B = if-B-then-not-A = Nand(A,B)
  • FFFT | Not-A-and-Not-B = Not(A-or-B) = not(if-not-A-then-B) = not(if-not-B-then-A) = Nor(A,B)
  • FFTF | Not-A-and-B = Not(A-or-Not-B) = not(if-B-then-A) = not(if-not-A-then-not-B)
  • FTFF | A-and-Not-B = Not(Not-A-or-B) = not(if-A-then-B) = not(if-not-B-then-not-A)
  • TFFF | A-and-B = Not(Not-A-or-Not-B) = not(if-A-then-not-B) = not(if-B-then-not-A)

Grounds if...then..., and, or, nand, nor

  • TTFF | A = (A or B) and (A or not-B)
  • TFTF | B
  • TFFT | A-iff-B = Not-A-iff-Not-B = (if-A-then-B) and (if-B-then-A) = (A and B) or (not-A and not-B) = not(A and not B) and not(not A and B)
  • FTTF | A-iff-Not-B = Not-A-iff-B = (if-A-then-not-B) and (if-B-then-not-A) = (A and not-B) or (not-A and B)
  • FTFT | Not-B
  • FFTT | Not-A

Grounds A, not-A, iff

Nand and Nor = supernegation = do nothing action

  • Nand outputs signal if any input is silent, and outputs silence if all inputs have signals.
  • Nor outputs silence if any input has signal, and outputs signal if all inputs have silence.

So apparently can express the halting problem: Nor expresses that the problem has halted.

  • Not-A = A nand A.
  • A and B = Not (A nand B) = (A nand B) nand (A nand B)
  • A or B = (A nand A) nand (B nand B)

A, B, C, each with value T or F, generate {$2^3$} input pairs TTT-TTF-TFT-TFF-FTT-FTF-FFT-FFF and yield {$2^{(2^3)}$} truth functions with outputs as follows:

  • TTTTTTTT Always true
  • FFFFFFFF Always false
  • TTTTTTTF A or B or C
  • TFFFFFFF A and B and C
  • FTTTTTTT Nand(A,B,C)
  • FFFFFFFT Nor(A,B,C)
  • TTTTTTFF A or B
  • TTFFFFFF A and B
  • TTTTTFFF
  • TTTTFFFF A
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This page was last changed on September 08, 2020, at 08:42 PM