DISCRETE MATH 3. On the Island of Knights and Knaves youu meet two natives, A an
ID: 3004056 • Letter: D
Question
DISCRETE MATH
3. On the Island of Knights and Knaves youu meet two natives, A and B. Given their statements below, what can you determine about their identities Fully explain your reasoning in words (as in my solutions in class and on the web and in Rosen's book Example 7, p. 19) (a) B says: "I am a knight only if A is a knight". What do you conclude? (b) B says: "I am a knight or A is a knight". What do you conclude? (c) In Case (b) above, what do you conclude if A adds: "I am a knave if and only if B is a knight"? L6ptsl xb send for S Send & Tra You have a free DoExplanation / Answer
(a) Step i: Define p and q as above.
Step ii: A said: p q.
Step iii: Observe that p and p q must have the same truth value.
Step iv: Construct a truth table for p q and compare its truth value to that of p:
pq
T
T
F
F
T
F
T
F
T
F
T
T
The truth values of p and p q coincide only in the case when p and q are both true. Hence
A and B must both be knights.
(b)
Step i: Define p and q as above.
Step ii: A said: pv q.
Step iii: Observe that p and p v q must have the same truth value.
Step iv: Construct a truth table for p v q and compare its truth value to that of p:
T
T
F
F
T
F
T
F
T
T
T
F
Observe that the truth values of p and pVq agree in rows 1, 3, and 4, we conclude that if A is a knight or B is a knight
(c)
Step i: Define p and q as above.
Step ii: A said: p <--> q.
Step iii: Observe that p and p <--> q must have the same truth value.
Step iv: Construct a truth table for p <--> q and compare its truth value to that of p:
T
T
F
F
T
F
T
F
T
F
F
T
Observe that the truth values of p and p <-->q agree in rows 1, 3 we conclude that if B is a knight then A is a knave
p qpq
T
T
F
F
T
F
T
F
T
F
T
T
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