# LMD Async Tutoring **URL:** \ **Category:** Tutoring\ **Created:** [April 23, 2024, 12:05am UTC](https://discuss.criticalfallibilism.com/t/lmd-async-tutoring/1933 "2024-04-23T00:05:39Z")\ **Posts on this page:** 1\ **Showing post:** 365
**Author:** ![LMD](https://discuss.criticalfallibilism.com/user_avatar/discuss.criticalfallibilism.com/lmd/32/3213_2.png) [@LMD](https://discuss.criticalfallibilism.com/u/LMD)\ **Post date:** [July 12, 2024, 4:12am UTC](https://discuss.criticalfallibilism.com/t/lmd-async-tutoring/1933/365 "2024-07-12T04:12:50Z")
> [@Eternity Async Tutoring](https://discuss.criticalfallibilism.com/t/eternity-async-tutoring/1932/297): > > ok can you figure out an expression which gives all the correct outputs for this table? | x | y | goal(x, y) | | --- | --- | --- | | 1 | 1 | 0 | | 1 | 0 | 1 | | 0 | 1 | 0 | | 0 | 0 | 0 | (These tables are just stock Obsidian tables copied and pasted from my note and they render perfectly. cool!) 3 false and 1 true input scenario. The true input scenario is: > (x ^ ~y) _Correct_ > [@Eternity Async Tutoring](https://discuss.criticalfallibilism.com/t/eternity-async-tutoring/1932/299): > > yes. ok now try to make an expression for this table. 10min max. | x | y | goal-2(x, y) | | --- | --- | --- | | 1 | 1 | 1 | | 1 | 0 | 1 | | 0 | 1 | 0 | | 0 | 0 | 0 | two true input scenarios: 1. (x^y) 2. (x^~y) so (x^y)v(x^~y) is an equivalent expression for goal-2(x, y) > goal-2(x, y) = (x^y)v(x^~y) also > goal-2(x, y) = (x) i.e just (x) is equivalent to goal-2(x, y) * * * > [@Eternity Async Tutoring](https://discuss.criticalfallibilism.com/t/eternity-async-tutoring/1932/302): > > ok what about an expression for: | x | y | goal-3(x, y) | | --- | --- | --- | | 1 | 1 | 1 | | 1 | 0 | 0 | | 0 | 1 | 1 | | 0 | 0 | 1 | So there are 3 true input cases. To make it simple, if I negate the false input case, it’ll imply the three true input cases. So: > goal-3(x, y) = ~(x^~y) I know I’m mixing up prefix and infix notation and that’s not ideal. * * * ### False input scenarios: so we know that: > goal-3(x, y) = ~(x^~y) What this means is: goal-3 is true when it’s not the case that x is true and y is false. So what is goal-3 when it _is_ the case that x is true and y is false? In that case, goal-3 is false: > ~goal-3(x, y) = (x^~y) This means what we just said which is that goal-3 is false when x is true and y is false. So the negation of the case equivalent to goal-3 being true, is the equivalent to the case of goal-3 being false. * * * > [@Eternity Async Tutoring](https://discuss.criticalfallibilism.com/t/eternity-async-tutoring/1932/312): > > What if every output needs to be false? okay so i.e what is the expression for: | x | y | goal-false(x, y) | | --- | --- | --- | | 1 | 1 | 0 | | 1 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 0 | you could specify and then negate all the false input scenarios: > goal-false(x, y) = ~((x^y)v(x^~y)v(~x^y)v(~x^~y)) ![Screenshot 2024-07-12 at 1.56.55 PM](https://discuss.criticalfallibilism.com/uploads/default/original/2X/7/77728f24aef5ff5bbaa0aff50afaea81fa358b9a.png) This means: goal-false(x, y) is true when there is an input scenario other than the four possible input scenarios, which since impossible is _never_, so it’s _always_ false. You could do simpler though: > goal-false(x, y) = ~(xv~x) ![Screenshot 2024-07-12 at 1.58.05 PM](https://discuss.criticalfallibilism.com/uploads/default/original/2X/f/fae7980fd73217d623f70a67ce58c90a279e6ab8.png) Which means: goal-false is true when it’s not the case that x is true or not x is true. That’ll mean that goal-false is never true because in all input scenarios x is either true or false. So ~(xv~x) always outputs false like goal-false(x, y). ~(yv~y) also works for the same reason. * * * I feel like I’ve made some good progress on my problem with understanding the concepts and I notice it feels great. Like I’ve really enjoyed studying and practising today more than other days and I’ve spent more time doing it over what I scheduled/expected. --- _[View the full topic](https://discuss.criticalfallibilism.com/t/lmd-async-tutoring/1933)._