how many different functions are there from a set with 10 elements to a set with the following numbers of elements:
a)2 b)3 c)4 d)5
To determine the number of different functions from a set with 10 elements to another set with a specific number of elements, we need to consider the cardinality of the two sets involved.
a) Set with 2 elements: In this case, for each element in the initial set, we have 2 choices to map it to one of the 2 elements in the target set. Therefore, the total number of different functions is 2^10 = 1024.
b) Set with 3 elements: Similarly, for each element in the initial set, we have 3 choices to map it to one of the 3 elements in the target set. Therefore, the total number of different functions is 3^10 = 59049.
c) Set with 4 elements: We have 4 choices for each element in the initial set. Therefore, the total number of different functions is 4^10 = 1048576.
d) Set with 5 elements: For each element in the initial set, we have 5 choices to map it to one of the 5 elements in the target set. Therefore, the total number of different functions is 5^10 = 9765625.
In summary:
a) 1024 different functions.
b) 59049 different functions.
c) 1048576 different functions.
d) 9765625 different functions.
To find the number of different functions from a set with 10 elements to a set with a specific number of elements, we need to use the concept of cardinality and the rule of mappings.
a) To find the number of different functions from a set with 10 elements to a set with 2 elements:
In this case, each element in the domain set (set with 10 elements) has two possible choices to be mapped to in the codomain set (set with 2 elements). For each element in the domain set, we have 2 choices. Since there are 10 elements, the total number of different functions would be 2^10 = 1024.
b) To find the number of different functions from a set with 10 elements to a set with 3 elements:
Similar to the previous case, each element in the domain set has three possible choices to be mapped to in the codomain set. Hence, for each element in the domain set, we have 3 choices. As there are 10 elements, the total number of different functions would be 3^10 = 59049.
c) To find the number of different functions from a set with 10 elements to a set with 4 elements:
Again, each element in the domain set has four possible choices to be mapped to in the codomain set. Therefore, for each element in the domain set, we have 4 choices. As there are 10 elements, the total number of different functions would be 4^10 = 1,048,576.
d) To find the number of different functions from a set with 10 elements to a set with 5 elements:
Similarly, each element in the domain set has five possible choices to be mapped to in the codomain set. So, for each element in the domain set, we have 5 choices. As there are 10 elements, the total number of different functions would be 5^10 = 9,765,625.
Therefore, the number of different functions from a set with 10 elements to sets with 2, 3, 4, and 5 elements are:
a) 1024 functions
b) 59049 functions
c) 1,048,576 functions
d) 9,765,625 functions