Answer to a) is n? and b) is n-1?How many functions are there from the set {1,2,...,n}, where n is a positive integer, to the set {0,1}
a) that assign 0 to both 1 and n?
b) that assign 1 to exactly one of the positive integers less than n?
a) There are no functions that assign 0 to both 1 and n.How many functions are there from the set {1,2,...,n}, where n is a positive integer, to the set {0,1}
a) that assign 0 to both 1 and n?
b) that assign 1 to exactly one of the positive integers less than n?
No function can have two pairs with the same first term: .
b) Notice that the range is
If the answer is
If the answer is
The wording is misleading. A function assigns 0 to 1 & n if the pairs (0,1) and (0,n) are in the function.
That is the reason for my comment.
So the statement of the question is incorrect.
As for your objection about the b) part, do you know that 0 is not a positive integer?
Of course I know that! But I solved it using the same "definition" as part (a). That is, for some . It can't be that since 0 is not in the domain of f!
I agree that the wording is ambiguous/incorrect... however this way it makes much more sense, more so because 0 is not in the domain.
I agree that a common terminology for communication is needed, however there are always exceptions... In my opinion, it is fairly obvious, (I really don't see any teacher giving such a question, in the way that you meant it) in this context, that f assigns 0 to both n and 1 means .
Yes, I agree that on a usual context it would not be correct to state it this way, however I see no point at all in asking this question if 0 is not even in the domain of f.
I guess we should just wait for zpwnchen to clarify the question.
I checked the question word by word from the textbook and it was correct as it was.
The solution manual said...How many functions are there from the set {1,2,....,n}, where n is a positive integer, to the set {0,1}
a) that assign 0 to both 1 and n?
b) that assign 1 to exactly one of the positive integers less than n?
a) is if n>1; 1 if n=1;
b) is 2(n-1)
But no idea of b answer...
My first question must be “Is the textbook written in English?”
If not then I have no basis for my objections.
But if it is written in English, then the author is abusing standard mathematical usage.
It should read:
a) assign or map both 1 and n to 0.
b) assigns or maps exactly one of the positive integer less than n to 1.
In standard mathematical usage saying assigns x to y means .
To answer the question correctly stated,
“How many functions assign exactly one of the positive integer less than n to 1”?
Answer
If there are none.
If , we can chose integers to map to 1.
The other must map to while itself may be assigned to either .
Therefore the answer is .