expr bit-wise "and" operator, dual of |
Arguments must be integers, result is an integer.
Bit n of the result is 1 if bit n of each argument is 1. Otherwise, bit n of the result is 0.
For negative arguments, we use the extended definition of ~ that ~$a==-1-$a. There are then the following cases:
| Case | Result |
|---|---|
| $a>=0,$b>=0 | Ordinary bitwise & |
| $a>=0,$b<0 | $a&$b == $a & ~(~$b) Contrapositive law $a&$b == $a & ~ (-1-$b) Extended definition of ~ Since -1-$b is positive, $a & ~(-1-$b) can be evaluated in bitwise fashion. |
| $a<0,$b>=0 | Commute to ($b & $a) and use the calculation above |
| $a<0,$b<0 | $a&$b == ~ (~$a | ~$b) De Morgan's Law $a&$b == ~ ((-1-$a) | (-1-$b)) Extended definition of ~ $a&$b == -1-((-1-$a) | (-1-$b)) Extended definition of ~ Since -1-$a and -1-$b are both positive, the expression ((-1-$a) | (-1-$b)) can be evaluated in the ordinary bitwise fashion. |
For logical/short-cut "and" use the && operator.