User:Jon AwbreyCactus Patch

Boolean Functions on Two Variables

Boolean Functions, Entitative Graphs, Existential Graphs

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\(\text{Boolean Functions on Two Variables}\)

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\(\text{Boolean Functions on Two Variables}\)

\(\text{Boolean Function}\)	\(\text{Entitative Graph}\)	\(\text{Existential Graph}\)
\(f_{0}\) \(\text{false}\)	32px \(\text{false}\)	32px \(\text{false}\)
\(f_{1}\) \(\text{neither}~ x ~\text{nor}~ y\)	32px \(\lnot (x \lor y)\)	64px \(\lnot x \land \lnot y\)
\(f_{2}\) \(y ~\text{and not}~ x\)	64px \(\lnot x \land y\)	40px \(\lnot x \land y\)
\(f_{3}\) \(\text{not}~ x\)	32px \(\lnot x\)	32px \(\lnot x\)
\(f_{4}\) \(x ~\text{and not}~ y\)	64px \(x \land \lnot y\)	40px \(x \land \lnot y\)
\(f_{5}\) \(\text{not}~ y\)	32px \(\lnot y\)	32px \(\lnot y\)
\(f_{6}\) \(x ~\text{not equal to}~ y\)	64px \(x \ne y\)	64px \(x \ne y\)
\(f_{7}\) \(\text{not both}~ x ~\text{and}~ y\)	64px \(\lnot x \lor \lnot y\)	32px \(\lnot (x \land y)\)
\(f_{8}\) \(x ~\text{and}~ y\)	64px \(x \land y\)	32px \(x \land y\)
\(f_{9}\) \(x ~\text{equal to}~ y\)	64px \(x = y\)	64px \(x = y\)
\(f_{10}\) \(y\)	32px \(y\)	32px \(y\)
\(f_{11}\) \(\text{if}~ x ~\text{then}~ y\)	40px \(x \Rightarrow y\)	64px \(x \Rightarrow y\)
\(f_{12}\) \(x\)	32px \(x\)	32px \(x\)
\(f_{13}\) \(\text{if}~ y ~\text{then}~ x\)	40px \(x \Leftarrow y\)	64px \(x \Leftarrow y\)
\(f_{14}\) \(x ~\text{or}~ y\)	32px \(x \lor y\)	64px \(x \lor y\)
\(f_{15}\) \(\text{true}\)	32px \(\text{true}\)	32px \(\text{true}\)

Boolean Functions, Truth Tables, Existential Graphs, Logical Formulas

\(\text{Propositional Forms on Two Variables}\)

\(\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\)
	\(x\colon\)	\(1~1~0~0\)
	\(y\colon\)	\(1~0~1~0\)
\(f_{0}\)	\(f_{0000}\)	\(0~0~0~0\)	\((~)\)	\(\text{false}\)	\(0\)
\(f_{1}\)	\(f_{0001}\)	\(0~0~0~1\)	\((x)(y)\)	\(\text{neither}~ x ~\text{nor}~ y\)	\(\lnot x \land \lnot y\)
\(f_{2}\)	\(f_{0010}\)	\(0~0~1~0\)	\((x)\ y\)	\(y ~\text{without}~ x\)	\(\lnot x \land y\)
\(f_{3}\)	\(f_{0011}\)	\(0~0~1~1\)	\((x)\)	\(\text{not}~ x\)	\(\lnot x\)
\(f_{4}\)	\(f_{0100}\)	\(0~1~0~0\)	\(x\ (y)\)	\(x ~\text{without}~ y\)	\(x \land \lnot y\)
\(f_{5}\)	\(f_{0101}\)	\(0~1~0~1\)	\((y)\)	\(\text{not}~ y\)	\(\lnot y\)
\(f_{6}\)	\(f_{0110}\)	\(0~1~1~0\)	\((x, y)\)	\(x ~\text{not equal to}~ y\)	\(x \ne y\)
\(f_{7}\)	\(f_{0111}\)	\(0~1~1~1\)	\((x\ y)\)	\(\text{not both}~ x ~\text{and}~ y\)	\(\lnot x \lor \lnot y\)
\(f_{8}\)	\(f_{1000}\)	\(1~0~0~0\)	\(x\ y\)	\(x ~\text{and}~ y\)	\(x \land y\)
\(f_{9}\)	\(f_{1001}\)	\(1~0~0~1\)	\(((x, y))\)	\(x ~\text{equal to}~ y\)	\(x = y\)
\(f_{10}\)	\(f_{1010}\)	\(1~0~1~0\)	\(y\)	\(y\)	\(y\)
\(f_{11}\)	\(f_{1011}\)	\(1~0~1~1\)	\((x\ (y))\)	\(\text{not}~ x ~\text{without}~ y\)	\(x \Rightarrow y\)
\(f_{12}\)	\(f_{1100}\)	\(1~1~0~0\)	\(x\)	\(x\)	\(x\)
\(f_{13}\)	\(f_{1101}\)	\(1~1~0~1\)	\(((x)\ y)\)	\(\text{not}~ y ~\text{without}~ x\)	\(x \Leftarrow y\)
\(f_{14}\)	\(f_{1110}\)	\(1~1~1~0\)	\(((x)(y))\)	\(x ~\text{or}~ y\)	\(x \lor y\)
\(f_{15}\)	\(f_{1111}\)	\(1~1~1~1\)	\(((~))\)	\(\text{true}\)	\(1\)