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Table 1 Logical matrices of 16 Boolean operations

From: State feedback control design for Boolean networks

Logical connective Logical operator Logical matrix for 2-node
  symbol Boolean networks
True \(\left (\begin {array}{cccc} 1&1&1&1 \\ 0&0&0&0 \end {array} \right)\)
False \(\left (\begin {array}{cccc} 0&0&0&0 \\ 1&1&1&1 \end {array} \right)\)
Proposition x 1 x 1 \(\left (\begin {array}{cccc} 1&1&0&0 \\ 0&0&1&1 \end {array} \right)\)
Proposition x 2 x 2 \(\left (\begin {array}{cccc} 1&0&1&0 \\ 0&1&0&1 \end {array} \right)\)
Negation (inhibition) x 1 ¬x 1 \(\left (\begin {array}{cccc} 0&0&1&1 \\ 1&1&0&0 \end {array} \right)\)
Negation (inhibition) x 2 ¬x 2 \(\left (\begin {array}{cccc} 0&1&0&1 \\ 1&0&1&0 \end {array} \right)\)
Conjunction \(\left (\begin {array}{cccc} 1&0&0&0 \\ 0&1&1&1 \end {array} \right)\)
Disjunction (union) \(\left (\begin {array}{cccc} 1&1&1&0 \\ 0&0&0&1 \end {array} \right)\)
Converse implication \(\left (\begin {array}{cccc} 1&1&0&1 \\ 0&0&1&0 \end {array} \right)\)
Material conditional \(\left (\begin {array}{cccc} 1&0&1&1 \\ 0&1&0&0 \end {array} \right)\)
Converse nonimplication \(\nleftarrow \) \(\left (\begin {array}{cccc} 0&0&1&0 \\ 1&1&0&1 \end {array} \right)\)
Material nonimplication \(\nrightarrow \) \(\left (\begin {array}{cccc} 0&1&0&0 \\ 1&0&1&1 \end {array} \right)\)
Biconditional \(\left (\begin {array}{cccc} 1&0&0&1 \\ 0&1&1&0 \end {array} \right)\)
Alternative denial \(\left (\begin {array}{cccc} 0&1&1&1 \\ 1&0&0&0 \end {array} \right)\)
Joint denial \(\left (\begin {array}{cccc} 0&0&0&1 \\ 1&1&1&0 \end {array} \right)\)
Exclusive disjunction \(\left (\begin {array}{cccc} 0&1&1&0 \\ 1&0&0&1 \end {array} \right)\)