Propositional Logic − Puzzle 1
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- Опубліковано 5 вер 2024
- Discrete Mathematics: Propositional Logic − Puzzle 1
Topics discussed:
On an island, there are two kinds of inhabitants, knights, who always tell the truth, and their opposites, knaves, who always lie. You encounter two people A and B. Determine, if possible, what A and b are if they address in the ways described.
(a) A says "B is a knight" and B says "The two of us are of opposite types".
(b) A says "At least one of us is a knave" and B says nothing.
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![Knights, Knaves, and Propositional Logic [Discrete Math Class]](/img/n.gif)
I am so confused 🤦.
Me too
Me too
Mee too
Me too
yes I can explain
just consider for case C) If you assume A-knight then according to him B is also knight on the other hand B is saying A-knave so it's contradiction so A can't be true and when you assume B-knight then that matches with the sayings of both as B is saying A-knave and A is saying both of us are knight which is false
so RESULT A-knave B-knight
case D)If you assume A-knight so according to him either I am knave OR B-knight but as we assumed A-knight he cant be knave so B-knight
but as some of the people in comments are saying both of them can be knave which is not true, if both of them are knave then the statement of A must be false which ic ~P OR Q but itwill never be false because of NOT P(NOT*FALSE=TRUE)
so RESULT A-KNIGHT B-KNIGHT
case e) is a funny case because you can't conclude anything, because it could be anything
1)A-KNIGHT B-KNIGHT
2)A-KNIGHT B-KNAVE
3)A-KNAVE B-KNIGHT
4)A-KNAVE B-KNAVE
here they are just saying about themselves and not for the other person so either they are saying the truth or false, GOD knows:")
@@VickySaha107 :)
@sayora isma don't worry let me explain, see in question no. C)
A says - Both of us are knight
B says - A is knave
Now see , let us first assume A is telling truth ok , so according to him B is also knight ok.
So if B is also knight then whatever he will say also be true, right
Now he says A is knave (SEE THE CONTRADICTION OCCURS FROM WHAT A SAID) SO our assumption of A BEING KNIGHT CANT BE POSSIBLE.
ASSUMPTION -2
A IS knave , now see he says he and B both are knight which can't be true because he is knave (INITIAL ASSUMPTION)
SO only two possibilities left - B can be knave just like A or B is knight and A is Knave okay.
So now if we assume B is also knave then he will always tell lies ,BUT SEE HE IS TELLING THAT A IS KNAVE (OUR INITIAL ASSUMPTION AND B IS TELLING) SO for sure B is not telling lies and he can't be KNave so only possibility left is A-Knave & B-Knight ( WHY? BECAUSE FROM WHAT WE ASSUMED FOR A , B IS ALSO LEADING US TO THE SAME TRUTH) ;) HOPE IT HELPS. IT NOT THEN EMAIL ME ILL SEND YOU THE SOLUTION ON PAPER.
@@vasubhatt6160 I dun understand about d I am sorry and thank if u can explain it to me
Thank u so much for your explaination
Thanks for your explanation but I am still confused in d
C. A is knave, B is knight
D. A is knight, B is knight
E. Can't say; any of the possibilities could be true
Thanks... Now Finally I can Complete First Question of My Assignment.
And Move On
C> A is knave , B is Knight.
D> 1) Both are knight.
2) Both are knave.
E> all four case like (00 01 10 11)
D both can never be knave. Knaves never say anything true, so if A is a knave, he can never say I'm a knave. Therefore, only first possibility is right, both are knights.
@@a.human. but if knaves can't tell anything true, then why knight can tell he is a knave which is that is a lie? we know that knight can never tell lie isn't that so?
@@muhammadhilwan7406 "A" being a Knight is not saying that he is Knave, he only ment either he is a Knave or B is a knight
As B is a Knight, he dont have to be a Knave validating his statement being Knight
My trick to solve the problem in the video
Part 1 -> Make a truth table like
A. Says ( B is a knight )
B. Says ( The two of us are opposite)
A B
K K. > False
K N. > False
N K. > False
N N. > True
Case 1 - if A is knight then because he always says the truth so B is also knight hence B will also say truth but B ..... Has said that we are opposite ....so...FALSE
Case 2 - if A is knight then because he always says the truth so .... **B is a knight** ... But 2nd row in truth table considers B to be a knave .... So ...FALSE
Case 3 - if A is a knave then he will always lie so B is a knave too then ( but 3rd row considers B to be a knight ) ....so ..FALSE
Case 4 - if A is a knave so he will lie that means B is a knave
( This matches with the what we consider for row 4 ) hence TRUE
I know it's complicated but do it row by row of truth table then it will be easy
Thank u so much
wait in the fourth case we are assuming both A and B to be a Knave ,
and if B is a Knave
that means he lies
which would mean
that A is a Knight
which would be a contradiction to our assumption
@@669_snehajitdey2 yes so if B is lying that means they are not opposite BUT THE SAME. which is true as both are considered naive.
@@sargun_narula oh my bad i was actually talking about the part c of the homework problem given in the video
Everyone can confuse 1st time.
I can watch the video and then I am also confused and i stop the study and say "faltu question".... But next day I go to study and say yourself "1bar try karte hain". And finally I know how to solve this questions and get relaxed...😌🛌
The answer is A is knave and B is also Knave.
c. A: knave, B: knight
d. A,B: knight
e. Both can be either knight or knave
I got the same ...is this the right one though??
@@arijitdassharma3005 i guess
even I am getting the same answer.
What are the propositional equations for c, D, ఈ problems
same , i am also getting dis
I think for option d . A, B: knight and A, B : knave both are true
(a) A is a knight => B is a knight => A is a knave (the opposite type). A contradiction. Therefore, A is a knave. Therefore, B is a knave (which works out, as they are then the same type).
(b) A is a knave=>both are knights=>A is a knight. A contradiction. So, A is a knight. => B is a knave. ☐
the opposite of "at least one of us is knave " can also be both are knave, and if both are knave the second proposition is true.
here we can come up with different inference that both are knave
C )
A-knave , B -Knight
D)
A-knight , B - Knight
E)
Any combination satisfy the condition
hey, can you explain those to me, i am stucked:(
@@ahsanulrahimsajitdipto499 there is a easy way to do this types of problems. First make 4 case like that truth table and try to analyse each case .
4 cases are
1)Knight knight
2)Knight knave
3)Knave knight
4)Knave knave
@@satyamkalyane6841 hi satyam gow can d be both are knight because if A is knight, he should say truth and tell he is knight but he is telling i am knave which can not be true
@@sakshiiikuthari1017A makes a statement using "or". For what A says to be true (speaking truth as a knight always does!) then his "or" statement has to be true. In prop logic (r or s) to be true either r is true or s is true or both are true. Since he is not a knave(we've assumed he's a knight") , then his second statement B is a knight must be true to make the total statement true. So it does work for A to be a knight, and B to be a knight. Perhaps this helps. Pardon my grammar mistakes.
👍
Sir please post correct answers for home work problems
c)A.knave,B:knight
d.both are either knave or knight
e.both are either knave or knight
D) no. both of them are only knight because is you find the truth value for considering both of them are knave, you can't get a false value for it and it always true so a cant be knave.
In d, it can be possible that both a and b are knave
@@vasubhatt6160 no the statement provided by A has to be false in order make him a knave so "i am knave" and "I am knight" both have to be false in order to make him a knave and if both of them are false then it make sense and B is a knave according to the statement of A as A is a knave and he lies
@@vasubhatt6160 right bro
For answer D:
A cannot be a knave . Why?
Ans: if he is a knave then his statement will always become true ("I am a knave" becomes true and "true or B is a knight" will always be true)which is contradiction.
Hence A will always be knight. if so then he must always say true .TO make his comment true B has to be Knight.
So final answer : A=Knight B=Knight.
For anybody in the future I got the same. remember de morgans law. !(PVQ) === !P ^ !Q.
Therefore, whenever A is supposed to be a knave, saying "I am not a knave" doesn't work logically. So anywhere where A is a knave is not possible.
for E the solution goes forth:
either A is a knight and B is a knight
A is a knave and B is a knave
A is a knight and B is a knave
A is a knave and B is knight
those are the possiblities to be tested at once
c) A is a knave, B is either Knight or Knave.
d) Either (A=Knight and B=Knight), Or (A=Knave and B=Knave).
e) A and B can be either Knight or Knave. Whatever they are, (e) is consistent.
Reasoning:
c) If P=F, then (P^Q) has to be F. That concludes for any value (T or F) of Q , (P^Q) is F. So P=F, Q=T or F.
d) If P=T, then (¬PvQ) has to be T which is only possible if Q=T. So, P=T and Q=T.
If P=F then (¬PvQ) has to be F which is only possible if Q=F. So, P=F and Q=F
e) If P=T, then P has to be a Knight, so P=T.
If P=F, then P has to be knave, so P=F
If Q=T, then Q has to be a Knight, so Q=T.
If Q=F, then Q has to be knave, so Q=F. Therefore, P and Q can be anything for (e) to be consistent.
in c if b is knave then he tells false means a is knight and if a is knight means he will tell truth means both are knight but b is knave so b cannot be knave.
in D, if P=F then (~p or q) can't become false, since ~p is true. How you can conclude both can be knaves
bro needs to study again
Great explanation ❤
thank you sir but where i will get the correct answers
(c) A is knave and B is knight
(d) A and B both are knight
(e) both can be knave or knight
@V Squad yes I can explain
just consider for case C) If you assume A-knight then according to him B is also knight on the other hand B is saying A-knight so it's contradiction so A can't be true and when you assume B-knight then that matches with the sayings of both as B is saying A-knave and A is saying both of us are knight which is false
socRESULT A-knave B-knight
case D)If you assume A-knight so according to him either I am knave OR B-knight but as we assumed A-knight he cants be knave so B-knight
but as some of the people in comments are saying both of them can be knave which is not true, if both of them are knave then the statement of A must be false which ic ~P OR Q but it never is false because of NOT P
so RESULT A-KNIGHT B-KNIGHT
case e) is a funny case because you can't conclude anything, because it could be anything
1)A-KNIGHT B-KNIGHT
2)A-KNIGHT B-KNAVE
3)A-KNAVE B-KNIGHT
4)A-KNAVE B-KNAVE
here they are just saying about themselves and not for the other person so either they are saying the truth or false, GOD knows:")
@@vasubhatt6160 😂
After one year
C: If A's statement is true, than B's statment is false, which is contradictory. Thus, A's statement is false, inturn making B's statement true.
A - Knave, B - Knight
D: The intended solution is likey that A being a Knave would make the statement true, hence they must be a Knight, which means the statement must be true, thus B must also be a Knight.
However, the question writer appears to have forgotten about exclusive or, where if both statements are true, then the or statement itself is false. Thus, a Knave could say "I am a Knave or B is a Knight", so long as the or is exclusive and B is a Knight. Without the assumption that the "or" is specifically INCLUSIVE or, there is more than one possible answer.
A - Either, B - Knight
E: There is no way to tell. Both A and B could be either a Knight or a Knave.
A - Either, B - Either
C) A knave and B knight
D) Both A and B are knight
E) Both A and B either knave or knight
How you solve question no. D
@@nishantlaxkar7451 or is there in condition of A , so either a is knave or b is a knight hence , second condition turns out to be true
In case E) all four cases are true because whatever A or B are they both are independent of each other.
Total cases: 1)A is a knight and B is a knave.
2)A is a knave and B is a knight.
3)Both A and B are knight.
4)Both A and B are knave.
Absolutely right
C:A is a knave B is a knight
D: A and B are both knights
E: Cannot be determined
brilliant
@@Shubham-hk6yf in case D aren't they actually both knaves? I think that s more of an exclusive OR than of an inclusive one. And they cannot both be knights if there s that " I am a knave" implying " p is false" if we assume that p is true. And if we take the inclusive OR, and say that A is a knight, then "i am a knave" means A is a knave, meaning that p is false, though we just assumed that p is true - and if that s the case, then "i am knave" becomes not P, which is true - then it wouldn' t matter if B is Knave or Knight, the statement would be true no matter what.
@@robertgabrielzaharie5405 bro your assumption is wrong. B cannot be a knave.q=F, then (~pvq) shuld be false
Thank you for the great explanation.
8:45 The expression could be "not p or not q" too. It eventually means at least one is knave. Right?
a certain kingdom, there were knights and knaves. knights always tell the truth and knaves tell always lies. there are two people, ed and ted. ed says "ted and i are different." ted claims, "only a knave would say that ed is a knave". in which category does each belong using truth table. please request you to explain clearly how both are knaves with truth table
Sir explain this too
ted is a knave and ed is a knight
Give its correct answer with explanation
C.A is knave; B is knight.
D. Both can be knave or knight.
E. Both can be knave or knight.
right bro
can you please explain this? How did you construct the logical expressions? Thanks
can you make mathematical propositional statements for c,d & e
@@akanvictor3271 In d and e you ll see there by proposition both can be knave and knight in all cases... In c if you take b as knight you would be able to solve further.
Right...bro
Both are knave and knight
I think for q 1
(C) A is knave b is knight
(D) Both are knave
1:45 B's statement is ambiguous because i thought "two of us are opposite types means either they both are Knight or Knave ....
it should have been written as B says " A is opposite types of Me " or " We are opposite types of each other " .
Sir ur explanations are nice
For D I've had to revisit. Hoping he goes over these in next lesson.
Seems like you liked the D
For (c)
B is knight
A is knave
For (d)
A can be either
B is knight
For (e)
A snd B both knight
A and B both knave
A can b knave or B knight
At right time for (e)
But A knight B knave not
you know you're fcked when you brain stops functioning mid way ....
thank you for the lecture sir :)
How did u got the logic p^notqand not p^q?
By two are opposite types
Explanation 👌🔥
I have a question in the statement
"A says 'at least one of us is knave' and B says nothing "
We can also conclude that both are knaves, in this A is lying that one of them is knaves as both of them are knaves and B is saying nothing and we don't need to think about that
I think that two conclusion can be drawn from the statement that
(I) As mentioned in Video that A is Knight and B is Knave
(II) Both A and B are knaves
A is not saying ‘one of them is knave’. It says ‘at least one of them is knave’. So, if both of them are knaves, A is true. But we said that A is lying. It is not possible
Still not understanding
@@ceyhunugur4631 thanks, got it now
you cant change in every case values of true or false it is given in the start and you need to make case for that values not take them out of your sleeve. ;)
P, P ->Q entailed P?
Is it true or false?
Mind blowing
is homework question e a tautology?
I love Neso Academy. ❤️
I thought to do this comment.
Well, that makes a lot of sense
What are the propositional expressions for C,D,E?
Hi
for statement D: its unsolvable because so long as we assume that knights always tell the truth and knaves always tell lies then.
let assume that A in that statement is a knave but " I am a knave or B is a knight" we notice here that "I" stands for "A" hence making a knave the truth teller which is not possible
okay case 2: lets assume A to be a knight that "I" again makes the knight a lier which is still not also possible hence unsolvable
Assume A is knight. A says " I am a knave or B is a knight".According to truth table of 'or' to make that sentence true at least one must be true so therefore I am knave is false then B is knight is true
Lastly A and B both are knight
But can you explain how to make compound proposition??
Someone Please can explain me the solution for e. as how is it converted to logical expression? and what logical operation makes all the values true? like disjunction, conjuction, is there any operation which makes all values true
Huh good question. However perhaps we don't need truth table. Either could be lying or telling the truth.
my brain: "ouh yeah eazy but how" lol xaxaaxa
Thank kiu
Sir please explain homework problems
for d both are knights
c) A is a knave, B is a knight.
d) Both A and B are knights.
e) Either of them can be a knave or a knight.
can u explain why A and B are not kneaves? if A is knight then why he said "i am kneave" which is false, and since he's false he must be kneave
Neso acamedy....can you upload the final answer?
We can find out the ans from case 1 itself
If A says B is knight then he is lying B is not knight,so B is knave and if B is knave then he is lying saying they r opposite types so they r same
C)B is knight and a is knave
Pinocchio is a wooden puppet whose nose grows longer whenever he tells a lie.
if he said: "my nose will grow right now".
will his nose grow or not? something to think about :)
Great explanation. He could teach it to a 5yr old
Finally understand
First of all Thank you very much for the video@Neso Academy.
I have a question though
Lets consider p=a is a knight,q= b is a night.
Now since A can either be 1)knight or 2)knave
case 1: A is knight
This wont hold since p exor q will be false.
case 2: A is a knave(not a night)
popositions are:a) !p
b)!q
c)p exnor q
This definitely holds. Hence a and b are knaves.
Is this right too?
This logical reasoning almost confusing
If A IS KNAVE , B IS ALSO KNAVE then B is also saying that they are of opposite types! How ? Then what's true?
Please help if anyone can explain this🙏
@@gourangageeksforgeeks
B is knave that's why whatever B says that has to be false,
means "the both are opposite types " is false so , they both are same types !!!! \
is it ok?????
@@gourangageeksforgeeks this simply means that both are wrong .
So, opposite of B's statement is "they are of same types "
and they both are knaves.
I hope you understands this correctly .
Home Work Answers:
C. A is Knave, B is Knight.
D. Both can be knave or knight.
E. Both can be knave or knight.
Your answer for D is wrong
I think for D. A and B are knights
Did you completed the playlist how was ur results.
Everyone can confuse??
c) A is knave that means P = F
(P ^ Q) must be false and it's confirmed.
B is knight because he says P = F
No contradiction
Both A and B says i am a knight
plz sir tell me
a says party time
Any one can tell logical expression to c,d and e
8:43
Sir , how to write the logical expression in this puzzle . I don't understand
same
Yes
Double quatation
Sir totally confused.please any explain it
d) A and B are both Knights
SOLUTIONS:
(PLEASE TELL ME IF I'M WRONG 😅)
C. 'A' IS KNAVE AND 'B' IS KNIGHT
D. I'm confused lol
E. Both can be KNIGHT but not knaves because it's mentioned that they are knights [F^F = F] Even if you take both of them knaves its going to be F for both when they say they're knights(as mentioned). Isn't it?
Don't take it like both are lieing so it's T.... BECAUSE F AND F IS FALSE!
Huh maybe you need to think about E again. We start with knowing nothing about either of them and whether they are knights or knaves. If they both say "I am a knight", they might both be telling the truth. We have no further information about them or their relationship with each other. So they might also both be lying or any combination - So could be knight knight, knight knave;, knave knight, knave knave. You did well on first one, good work. For D it's definitely trickier. Take it one step at a time. Remember that anything a knight says has to be true. So if it's (a or b) that a knight says, then either a is true, b is true or both are true. And if it's (a or b) that a knave says, then (a or b) has be false because the knave always lies. Hoping this info is helpful. Logic is basis of all clear thinking and math, science, and reflection on one's own behavior. Lot to learn.
IS it only my brain which is dumb or everyone else
I AM ALSO DUMB DONT BE PANIC
Knights knaves p, q 🤯🤯confused😵😵
OMG 🥴 so confusing 🥴🥴🙄
What is going on there 😱😳🙄🥴😭
😂😂
After watching half of the video, i couldn't understand anything... So i paused the video and started going by traditional method of guessing and it was soon solved.💀
I am not able to solve the puzzle properly
Same here....
Answers of HW:
(c) A is Knave and B is Knight
(d) A is Knight and B is Knight or A is Knave and B is Knave
(e) A can be anyone and B can be anyone
just do it by tabular method...it is too confusing
Still not understanding
A merko chakkar aara hai 🤒
same bhai
Still confuse
these knev knight are not uns=derstandable to me 🤤🤤🤤🤤🤤
I am confused
It's so confusing
Sir please pin the right answer
Sir please pin the right answer in comment section
This problem makes me laugh.
this is the worst example anyone could use
das sprehen zu schnell
wir das class ist super
confused
Very confusing😔😥
What was this😁😂