Propositional Logic − Puzzle 1

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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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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:")

vasubhatt

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

jayxcoder

Thanks... Now Finally I can Complete First Question of My Assignment.
And Move On

MDgaming

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)

abd_gaming

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

sargun_narula

c. A: knave, B: knight
d. A, B: knight
e. Both can be either knight or knave

joseph

C )
A-knave, B -Knight
D)
A-knight, B - Knight
E)
Any combination satisfy the condition

satyamkalyane

(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. ☐

nnaammuuss

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

dzenathan

c)A.knave, B:knight
d.both are either knave or knight
e.both are either knave or knight

yahyairfan

It is indeed confusing at first but lemme explain a trick here... we simply need to split the original proposition into 4 cases and solve those four cases just to see which one is true. The knight and knave case, consider this way: {(CASE-1) let A be knight (CASE-2) let A be knave (CASE-3) let B be knight (CASE-3) let B be knave}

ha_

Sir please post correct answers for home work problems

saivamsi

(c) A is knave and B is knight
(d) A and B both are knight
(e) both can be knave or knight

Atulkumar-blsv

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...😌🛌

KiranShirkey

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.

meetjoshi

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

JatinKumar-wf

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.

sadmanmohammadnasif

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

PowerStar

thank you sir but where i will get the correct answers

farhanakhtar

8:45 The expression could be "not p or not q" too. It eventually means at least one is knave. Right?

marco_robert

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