Lalph Rauren was famous for discovering potential models as he walked the streets of New York. One day he spotted four girls who later became top models: Amy, Mary, Kay, and Beth. One was discovered walking through an apartment building, one in a coffee shop, one in a shopping mall, and one in the library. You know the following.
a.If Kay was not discovered in the library, then Amy was discovered in the apartment building.
b.Beth was discovered in the library, unless Mary was discovered in the coffee shop.
c.If Amy was discovered in the apartment building, then Mary was discovered in the shopping mall.
d.If Beth was not discovered in the library, then neither was Kay.
If you have enough information from the statements above, match the girl with the location in which she was discovered and explain your reasoning. If you cannot make the match, explain why.
No one has answered this question yet.
Can it even be solved?
We can solve this problem in many ways, a truth table, or a logic tree. The latter seems more convenient, so it will be used.
First let's use some definitions of symbols.
For the names,
A=Amy
B=Beth
K=Kay
M=Mary
For the locations,
A=Apartment building
C=Coffee Shop
S=Shopping Mall
L=Library
Now the operators:
K=L means Kay was discovered in the library (etc.)
K≠L means Kay was not discovered in the library.
-> means "then", or "it follows".
and finally,
&xor; means exclusive or, exactly one of the two statements is true.
In logic, if the condition of a conditional statement is true, "it follows" that the following statement
is true. If the condition is NOT satisfied, we do not know if the following statement is true or not, i.e.
a->b
if a is true, we know that b is true
if a is not true, we don't know if b is true.
"If it rains, I stay home."
If it doesn't rain, I may stay home, or I may go out.
Now let's get started.
We will translate the four given statements:
1. K≠L ->A=A
2. B=L &xor; M=S
3. A=A -> M=S
4. B≠L -> K≠L
We have two cases, either (A) K was found in the Library, or (B) she was not.
case (A)
K=L
A=A (from 1.)
B≠L (since K=L)
But from 4, B≠L -> K≠L
Contradiction, therefore K≠L
Case (B)
K≠L
A=A (from 1, K≠L -> A=A)
M=S (from 3 A=A -> M=S)
B=L
(if(B≠L, then K≠L, then nobody was found in the library; => B=L)
K=C (by elimination)
Conclusion:
K=C,B=L,M=S,A=A
If you have been following the logic, you would be able to translate the conclusion.
Solve the following inequality. Then place the correct answer in the box provided. Answer in terms of an improper fraction.
3y + 5 >10
To solve this problem, we need to analyze the given statements and use deductive reasoning to make the match between the girls and the locations they were discovered in.
Let's break down the statements one by one:
a. If Kay was not discovered in the library, then Amy was discovered in the apartment building.
This statement tells us that if Kay was not discovered in the library, i.e., Kay is not in the library, then Amy was discovered in the apartment building. Therefore, if Kay is not in the library, Amy must be in the apartment building.
b. Beth was discovered in the library, unless Mary was discovered in the coffee shop.
This statement gives us two possibilities: Either Beth is in the library, or Mary is in the coffee shop. It can't be both because "unless" implies that only one of these two conditions can be true. So we have two potential pairs: (Beth, library) or (Mary, coffee shop).
c. If Amy was discovered in the apartment building, then Mary was discovered in the shopping mall.
This statement creates a conditional relationship between Amy being discovered in the apartment building and Mary being discovered in the shopping mall. If Amy is in the apartment building, then Mary must be in the shopping mall.
d. If Beth was not discovered in the library, then neither was Kay.
This statement tells us that if Beth is not in the library, then Kay is also not in the library. In other words, if Beth is not in the library, Kay cannot be in the library.
Based on these statements, let's analyze the possible scenarios:
1. If Beth is in the library: According to statement b, if Beth is in the library, Mary cannot be in the coffee shop. So Beth must be in the library, and Mary cannot be in the coffee shop. Now, according to statement d, if Beth is in the library, Kay cannot be in the library. Therefore, Kay cannot be in the library, and Amy cannot be in the apartment building (due to statement a). This leaves us with only one option: Amy must be in the shopping mall. So the match is: (Beth, library), (Mary, coffee shop), (Amy, shopping mall). This leaves Kay to be discovered in the apartment building.
2. If Mary is in the coffee shop: According to statement b, if Mary is in the coffee shop, Beth cannot be in the library. So Mary must be in the coffee shop, and Beth cannot be in the library. Now, according to statement d, if Beth is not in the library, Kay cannot be in the library. Therefore, Kay cannot be in the library, and Amy cannot be in the apartment building (due to statement a). This leaves us with only one option: Amy must be in the shopping mall. So the match is: (Mary, coffee shop), (Amy, shopping mall). This leaves Beth to be discovered in the library and Kay to be discovered in the apartment building.
Based on the analysis above, we can make the match as follows:
1. Beth was discovered in the library.
2. Mary was discovered in the coffee shop.
3. Amy was discovered in the shopping mall.
4. Kay was discovered in the apartment building.
This solution satisfies all the given conditions and makes a match between the girls and the locations they were discovered in.