Colours?
Selena, Jennifer and Miley wear a blue dress, yellow dress, and green dress in an unknown order. It is known that:
1) If Selena wears blue, then Jennifer wears green.
2) If Selena wears yellow, then Miley wears green.
3) If Jennifer does not wear yellow, then Miley wears blue.
What is the color of the dress Selena is wearing?
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9 solutions
i see each statement contrasts the other one and i got ur point but still confused ,all because condition 3 so please explain me how come ?
Problem being that the question has made no statement about what any of them are wearing in the current instance, and so it cannot be assumed.
I assumed that if Jennifer wore yellow, then Miley will not wear blue.
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False P will accept whatever Q and the statement of "if P, then Q" would still hold true.
TRUTH TABLE FOR IF P THEN Q
| P : not yellow J | Q : blue M | IF P, THEN Q |
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
In this case, P is false since J does wear yellow but Q is true because M still wears blue, and according to the second last row of the table, the statement "If Jennifer does not wear yellow, then Miley wears blue" is true anyway.
Your interpretation is suitable for a biconditional statement, but is wrong on a conditional one.
Biconditional statement : contains words of "IF AND ONLY IF" or contracted into "IFF". The original conditional statement we have here in this problem will turn into "Jennifer does not wear yellow IF AND ONLY IF Miley wears blue". When this newly crafted statement above is true, we can only have :
1) green Jennifer and blue Miley ✓
2) yellow Jennifer and green Miley ✓
for 2 possible final results of
1) green Jennifer, blue Miley and yellow Selena ✓ OR
2) yellow Jennifer, green Miley and blue Selena ✓
regardless of the first two clues / rules.
Biconditional : not yellow J <==> blue M
In biconditional, both part of the pair limited and controlled each other (that's the BI working).
Conditional statement : contains words of "IF" & "THEN" in the structural "IF (a condition / antecedent / protasis), THEN (a consequent)". The original conditional statement we have here is "IF Jennifer does not wear yellow, THEN Miley wears blue". When this original conditional statement above is true, we can have :
1) green Jennifer and blue Miley ✓
2) yellow Jennifer and green Miley ✓
3) yellow Jennifer and blue Miley ✓
for 3 possible final results of
1) green Jennifer, blue Miley and yellow Selena ✓ OR
2) yellow Jennifer, green Miley and blue Selena ✓ OR
3) yellow Jennifer, blue Miley and green Selena ✓
regardless of the first two clues / rules.
Conditional : not yellow J ==> blue M
In conditional, only one part of the pair limited and controlled the other.
I still don't get it ( AT ALL).
but that contradicts the 3rd statement. miley can only wear blue if jennifer isn’t wearing yellow. in this case jennifer has to wear green but selina is already wearing it.
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False P will accept whatever Q and the statement of "if P, then Q" would still hold true.
TRUTH TABLE FOR IF P THEN Q
| P : not yellow J | Q : blue M | IF P, THEN Q |
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
In this case, P is false since J does wear yellow but Q is true because M still wears blue, and according to the second last row of the table, the statement "If Jennifer does not wear yellow, then Miley wears blue" is true anyway.
From the final condition, we know that if Jennifer wears Blue or Green, Miley wears Blue. which means Jennifer wears green and Selena is left with yellow.
But, from condition one if Selena wears yellow Miley should wear Green. So this combination is eliminated and so we assume Jennifer wears yellow, the Selena is left with green and blue, if she wear blue the condition 1 will be violated and so wears green.
CAN'T SAY is the answer.
Given ==> Assumed based on elimination- that the 3rd must be a diff color
- 1. S(b) --> J(g) ==> M(y)
- 2. S(y) --> M(g) ==> J(b)
- 3. J(~y) --> M(b) ==> [J(g) -->M(b)] -->S(y)
in #3, the assumption that J(b) cannot exist, since it will automatically become M(b) by given #3, and both J and M are wearing blue which violates the rule of diff colors. Therefore in #3, only J(g) can be true.
from the three given statements above we can draw the cases for Selena's color:
start with: if S(b) then J(g) and M(y) , using #1 above. But when J(g), then M(b) also by condition #3. Therefore Miley conflicts , wearing both blue and yellow if you start with Selena wearing blue.
now if S(y) then M(g) and J(b) , using #2 above. But this J(b) is also a contradiction since J(b) cannot exist as stated in condition #3 above.
now we are left with either these two cases of Selena wearing green and the other two are either blue or yellow.
- S(g) --> [J(b)-->M(y)]
- S(g) --> [J(y)-->M(b)]
The first case is invalid because it has J(b) which cannot exist, with the reason stated in condition #3 above.
The second case has M(b), but it was implied in #3 that to have an M(b), J cannot wear yellow, which is clearly done here. So this case is invalid also.
There are no cases valid with the Given Conditions in the problem, so Selena has no choice but to go topless! :D
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I agree with your argument that if S(b) and S(y) will both lead to contradictions. I also agree that S(g) - J(b) - M(y) violates the conditions.
However, there are no issues with with S(b) - J(y) - M(b). It is not "implied in #3" that "If Jennifer wears yellow, then Miley cannot wear blue".
You are most likely thinking of the contrapositive. The contrapositive of #3 is actually "If Miley does not wear blue, then Jennifer does not does not year yellow", which means that "If Miley does not wear blue, then Jennifer must wear yellow".
Let me direct you to the Logic Wiki : Contrapositive : A statement is logically equivalent to its contrapositive. The contrapositive negates both terms in an implication and switches their position, so for example the contrapositive of P ⟹ Q is ¬ Q ⟹ ¬ P ( ¬ means "not" ).
A common misconception (similar to what you did) is to claim that P ⟹ Q implies that = P ⟹ = Q , which is not true. For example, "On weekdays, Calvin wakes up at 6am" does not imply that I cannot wake up at 6am on Saturday.
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The statement "it is known that:" creates a condition that all 3 conditions MUST be true. To violate any of the conditions makes the problem a fallacy. Re-word the problem statement.
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@Keith Bowers – Indeed, all 3 conditions are true. This is satisfied when Selena wears green, Jennifer wears yellow and Miley wears blue.
1) If Selena wears blue, then Jennifer wears green.
2) If Selena wears yellow, then Miley wears green.
3) If Jennifer does not wear yellow, then Miley wears blue.
1) Is trivially true since Selena does not wear blue.
2) is trivially true since Selena does not year yellow
3) is trivially true since Jennier wears yellow.
My point is that just because you are given a case of "If not P then Q", then it doesn't imply that "If P then not Q". For example, consider the statement "If Donald Trump is not the president, then the sun rises in the East", which is of course a true statement. If does not imply that "If Donald Trump is the presiden, then the sun does not rise in the East".
i like ur explanation
Scenario 1: Selena wears blue. If Selena wears blue, Jennifer wears green. If Jennifer does not wear yellow (ie. if she wears green), Miley wears blue. Selena and Miley cannot both wear blue, so this scenario doesn't work.
Scenario 2: Selena wears yellow. If Selena wears yellow, Miley wears green. But Jennifer isn't wearing yellow, so Miley has to wear blue. Miley cannot wear two colours at once, so this scenario also doesn't work.
Scenario 3: Selena wears green. If Selena wears green, Jennifer can wear yellow, forcing Miley to wear blue. It all works out, therefore Selena is wearing green.
1st clue implied a trio of blue S, green J and yellow M.
2nd clue implied a trio of yellow S, green M and blue J.
3rd clue implied a duo of green J and blue M (and thus, forcing a yellow S), but this is contradictory to both of the other clues.
Therefore, Selena is wearing neither a blue nor a yellow dress, but a green one.
Additional information
Jennifer wears a yellow dress and Miley wears a blue dress.
1) If S=B => J=G
2) If S=Y => M=G
3) If J = Y => M=B
1st Hyp:
(S=Y) : by 2 (M=G) and (J=B) : by 3 (M=B) Impossible because (M=G)
2nd Hyp:
(S=B) : by 1 (J=G) : by 3 (M=B) Impossible because (S=B)
3nd Hyp:
(S=G) : (J=Y): (M=B) Possible (Nothing says that if J=Y M cannot wear B)
(S=G) : (J=B): by 3 (M=Y) Possible
So Selena wears Green.
"(S=G) : (J=B): by 3 (M=Y) Possible"
The above is the only part that needs correction. Blue J and yellow M is impossible by 3.
The first and second IF statements about the color of Selena's dress both contradicted the third. The only other scenario not given was IF Selena wore the Green dress, then Jennifer would wear the yellow dress. The third IF statement only is activated in the event of Jennifer NOT wearing the yellow dress, otherwise, Miley has the choice to still wear the blue dress, since it is the only one left anyway.
The key for solving it is to write down all true prepositions about the world of the puzzle and find assumptions that do not contradict them.
- S ( b ) = > J ( g )
- S ( y ) = > M ( g )
- J ( ¬ y ) = > M ( b )
- Each person can wear one of b , g , y and any two people can wear same colour
We can allow ourselves to assume the colour of a dress Selena wears across all available colours, and see if the world allows this assumption to pass.
Assume S ( b ) . By №1 & №2: S ( b ) = > J ( g ) = > M ( b ) which contradicts №4. S ( b ) is false.
Assume S ( y ) . By №2 S ( y ) = > M ( g ) . However, together with №4 this implies J ( b ) . This, taken together with№3 J ( b ) = > M ( b ) and we get a contradiction with №4. S ( y ) is false.
Assume S ( g ) . Looking at two remaining colours, assume J ( b ) . However by №3 J ( b ) = > M ( b ) which leads to contradiction with №4. However, if we assume J ( y ) then no rule kicks in and assumptions S ( g ) , J ( y ) hold in the given world. By №4, then, M ( b ) .
The answer is S ( g ) .
(Sx refers to statement x)
If S->B then J->G (S2) then M->Y (S1) but M->B (S4), so S≠B
If S->Y then M->G (S3) then J->B (S1) but M->B (S4), so S≠Y
Assume S->G
If M->Y then J->B (S1) but M->B (S4), so M≠Y
M->B then J->Y (S1), and this is not countered by any statements. Thus, Green is the answer.
For those thinking S4 counters the last one: The statement says
"if Jennifer does not wear yellow..."
Jennifer is wearing yellow, so it is irrelevant as to what follows. It is a if statement, not a if and only if, so the then doesn't matter if the if isn't true.
Assume Selena wears each colour and find contradictions. If her wearing a certain colour leads to a contradiction, then she cannot be wearing the colour. a) Assume Selena wears blue. By proposition 1, Jennifer wears green. By proposition 3, Miley wears blue. But Selena is wearing blue by our assumption. This is a contradiction. Selena cannot be wearing blue. b) Assume Selena wears yellow. By proposition 2, Miley wears green. By proposition 3, Miley wears blue. This is a contradiction. Selena cannot be wearing yellow. c) Selena must be wearing green. By proposition 3, if Jennifer wears blue (not yellow), Miley wears blue, this is a contradiction. If Jennifer wears yellow and Miley wears blue there is no contradiction. Therefore Selena is wearing the green dress.
From condition 1, if Selena wears blue, then Jennifer wears green. Then, from condition 3, Jennifer is not wearing yellow so Miley is wearing blue. However, that is impossible since Selena is wearing blue! Thus, Selena can't be wearing blue.
From condition 2, if Selena wears yellow, then Miley wears green. This leaves Jennifer with blue. Then, from condition 3, Jennifer is not wearing yellow so Miley is wearing blue. However, that is impossible since Miley is wearing green! Thus, Selena can't be wearing yellow.
This leaves us with Selena wearing green, which doesn't violate any of the conditions (so long as Jennifer is wearing yellow and Miley is wearing blue).
Remark: There seems to be some confusion that Jennifer wearing yellow and Miley wearing blue would violate condition 3. However, this is not the case. The statement only speaks to what happens if Jennifer does not wear yellow. You can learn more about logical statements and contrapositives in Propositional Logic Word Problems .