hello, what is that meaing of that? :)
- Thread starter remy
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How many distinct functions f:{1,2}→{3,4,5} are there for specific range size.
in your case the range size is not given
here is similar example:
How many distinct functions f:{1,2,3,4,5}→{1,2,3} are there, from the set {1,2,3,4,5} to the set {1,2,3}, whose range is a set of size exactly 2?
You can choose a two element set from {1,2,3} in C(3,2)=3 ways, and for each such choice there exists 2^5 many functions. But 2 of them are constant. Thus, for each two element set of the range, there are 2^5-2 number of non constant functions. So, the total number of required functions are 3×(2^5-2)=90.
in your case the range size is not given
here is similar example:
How many distinct functions f:{1,2,3,4,5}→{1,2,3} are there, from the set {1,2,3,4,5} to the set {1,2,3}, whose range is a set of size exactly 2?
You can choose a two element set from {1,2,3} in C(3,2)=3 ways, and for each such choice there exists 2^5 many functions. But 2 of them are constant. Thus, for each two element set of the range, there are 2^5-2 number of non constant functions. So, the total number of required functions are 3×(2^5-2)=90.
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Is this precalculus? College algebra?
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College algebra
If A and B are finite sets with m and n elements, then there are exactly n^m functions from A to B. This is because each of the m elements can be mapped in n ways independently of each other. Hence there are n×n×…..n (m times product) = n^m such functions.
If A and B are finite sets with m and n elements, then there are exactly n^m functions from A to B. This is because each of the m elements can be mapped in n ways independently of each other. Hence there are n×n×…..n (m times product) = n^m such functions.
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In that case, this is not advanced math.
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f: {1, 2}-> {3, 4, 5} means that f is a function with domain {1, 2} and range {3, 4, 5}. That means that in y= f(x), x can be either 1 or 2 and y can be any of 3, 4, or 5. As Mathlover1 said, there are 3^2= 9 such functions. It is not difficult to list them all
1) f(1)= 3, f(2)= 3
2) f(1)= 3, f(2)= 4.
3) f(1)= 3, f(2)= 5.
4) f(1)= 4, f(2)= 3.
5) f(1)= 4, f(2)= 4.
6) f(1)= 4, f(2)= 5.
7) f(1)= 5, f(2)= 3.
8) f(1)= 5, f(2)= 4.
9) f(1)= 5, f(2)= 5.
1) f(1)= 3, f(2)= 3
2) f(1)= 3, f(2)= 4.
3) f(1)= 3, f(2)= 5.
4) f(1)= 4, f(2)= 3.
5) f(1)= 4, f(2)= 4.
6) f(1)= 4, f(2)= 5.
7) f(1)= 5, f(2)= 3.
8) f(1)= 5, f(2)= 4.
9) f(1)= 5, f(2)= 5.
