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Boolean Functions and Logical Graphs on Two Variables

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\text{Boolean Functions and Logical Graphs on Two Variables}

Boolean Functions and Logical Graphs on Two Variables

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Boolean Functions and Logical Graphs on Two Variables
\text{Boolean Function} \text{Entitative Graph} \text{Existential Graph}
f₀(x,y) Cactus Root
 
Cactus Stem
 
f_{0} \texttt{ } \texttt{( )}
f₁(x,y) Cactus (xy)
 
Cactus (x)(y)
 
f_{1} \texttt{(} x y \texttt{)} \texttt{(} x \texttt{)(} y \texttt{)}
f₂(x,y) Cactus (x(y))
 
Cactus (x)y
 
f_{2} \texttt{(} x \texttt{(} y \texttt{))} \texttt{(} x \texttt{)} y
f₃(x,y) Cactus (x)
 
Cactus (x)
 
f_{3} \texttt{(} x \texttt{)} \texttt{(} x \texttt{)}
f₄(x,y) Cactus ((x)y)
 
Cactus x(y)
 
f_{4} \texttt{(} y \texttt{(} x \texttt{))} \texttt{(} y \texttt{)} x
f₅(x,y) Cactus (y)
 
Cactus (y)
 
f_{5} \texttt{(} y \texttt{)} \texttt{(} y \texttt{)}
f₆(x,y) Cactus ((x,y))
 
Cactus (x,y)
 
f_{6} \texttt{((} x \texttt{,} y \texttt{))} \texttt{(} x \texttt{,} y \texttt{)}
f₇(x,y) Cactus (x)(y)
 
Cactus (xy)
 
f_{7} \texttt{(} x \texttt{)(} y \texttt{)} \texttt{(} x y \texttt{)}
f₈(x,y) Cactus ((x)(y))
 
Cactus xy
 
f_{8} \texttt{((} x \texttt{)(} y \texttt{))} x y
f₉(x,y) Cactus (x,y)
 
Cactus ((x,y))
 
f_{9} \texttt{(} x \texttt{,} y \texttt{)} \texttt{((} x \texttt{,} y \texttt{))}
f₁₀(x,y) Cactus y
 
Cactus y
 
f_{10} y y
f₁₁(x,y) Cactus (x)y
 
Cactus (x(y))
 
f_{11} \texttt{(} x \texttt{)} y \texttt{(} x \texttt{(} y \texttt{))}
f₁₂(x,y) Cactus x
 
Cactus x
 
f_{12} x x
f₁₃(x,y) Cactus x(y)
 
Cactus ((x)y)
 
f_{13} \texttt{(} y \texttt{)} x \texttt{(} y \texttt{(} x \texttt{))}
f₁₄(x,y) Cactus xy
 
Cactus ((x)(y))
 
f_{14} x y \texttt{((} x \texttt{)(} y \texttt{))}
f₁₅(x,y) Cactus Stem
 
Cactus Root
 
f_{15} \texttt{( )} \texttt{ }

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\text{Boolean Functions and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Orbit Order}

Boolean Functions and Logical Graphs on Two Variables

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Boolean Functions and Logical Graphs on Two Variables • Orbit Order
\text{Boolean Function} \text{Entitative Graph} \text{Existential Graph}
f₁₂(x,y) Cactus x
 
Cactus x
 
f_{12} x x
f₁₀(x,y) Cactus y
 
Cactus y
 
f_{10} y y
f₃(x,y) Cactus (x)
 
Cactus (x)
 
f_{3} \texttt{(} x \texttt{)} \texttt{(} x \texttt{)}
f₅(x,y) Cactus (y)
 
Cactus (y)
 
f_{5} \texttt{(} y \texttt{)} \texttt{(} y \texttt{)}
f₀(x,y) Cactus Root
 
Cactus Stem
 
f_{0} \texttt{ } \texttt{( )}
f₁₅(x,y) Cactus Stem
 
Cactus Root
 
f_{15} \texttt{( )} \texttt{ }
f₁(x,y) Cactus (xy)
 
Cactus (x)(y)
 
f_{1} \texttt{(} x y \texttt{)} \texttt{(} x \texttt{)(} y \texttt{)}
f₇(x,y) Cactus (x)(y)
 
Cactus (xy)
 
f_{7} \texttt{(} x \texttt{)(} y \texttt{)} \texttt{(} x y \texttt{)}
f₂(x,y) Cactus (x(y))
 
Cactus (x)y
 
f_{2} \texttt{(} x \texttt{(} y \texttt{))} \texttt{(} x \texttt{)} y
f₁₁(x,y) Cactus (x)y
 
Cactus (x(y))
 
f_{11} \texttt{(} x \texttt{)} y \texttt{(} x \texttt{(} y \texttt{))}
f₄(x,y) Cactus ((x)y)
 
Cactus x(y)
 
f_{4} \texttt{(} y \texttt{(} x \texttt{))} \texttt{(} y \texttt{)} x
f₁₃(x,y) Cactus x(y)
 
Cactus ((x)y)
 
f_{13} \texttt{(} y \texttt{)} x \texttt{(} y \texttt{(} x \texttt{))}
f₈(x,y) Cactus ((x)(y))
 
Cactus xy
 
f_{8} \texttt{((} x \texttt{)(} y \texttt{))} x y
f₁₄(x,y) Cactus xy
 
Cactus ((x)(y))
 
f_{14} x y \texttt{((} x \texttt{)(} y \texttt{))}
f₆(x,y) Cactus ((x,y))
 
Cactus (x,y)
 
f_{6} \texttt{((} x \texttt{,} y \texttt{))} \texttt{(} x \texttt{,} y \texttt{)}
f₉(x,y) Cactus (x,y)
 
Cactus ((x,y))
 
f_{9} \texttt{(} x \texttt{,} y \texttt{)} \texttt{((} x \texttt{,} y \texttt{))}