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NOT
Inverts 1 and 0.
AND
Takes two bits as inputs, only gives out 1 if both inputs are 1, 0 in all other cases.
OR
Takes two bits as inputs, gives out 1 if one of both inputs are 1, 0 only if both inputs are 0.
NAND
AND gate followed by a NOT gate, so all results of AND are inverted. Gives out 1 in all cases except when both inputs are 1.
NOR
OR gate followed by a NOT gate, so all results of OR are inverted. Gives out 0 in all cases except when both inputs are 0
XOR
Outputs 1 when exactly one of its inputs is 1. Two 0 or two 1 give 0.
XNOR
Inverted XOR gate, outputs 0 when exactly one of its inputs is 1. Two 0 or two 1 give 1.