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How many onto functions can be defined from the set A = {1, 2, 3, 4} to {a, b, c}?
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How many onto functions can be defined from the set A = {1, 2, 3, 4} t...
An onto function is a function in which every element in the range (in this case, the set {a, b, c}) is mapped to by at least one element in the domain (in this case, the set {1, 2, 3, 4}). There are three elements in the range, so each of them must be mapped to by at least one element in the domain.
There are 4 elements in the domain, so there are 4 possible choices for the first element in the range, 3 remaining choices for the second element, 2 remaining choices for the third element, and 1 remaining choice for the fourth element. This means there are a total of 4 x 3 x 2 x 1 = 24 possible onto functions that can be defined from the set {1, 2, 3, 4} to {a, b, c}.

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How many onto functions can be defined from the set A = {1, 2, 3, 4} t...
Solution:

To find the number of onto functions from A to {a,b,c}, we need to consider the following cases:

Case 1: All four elements of A are mapped onto different elements of {a,b,c}

In this case, the first element of A can be mapped onto any of the three elements of {a,b,c}. Similarly, the second element of A can be mapped onto any of the remaining two elements of {a,b,c}. Continuing this way, the third element of A can be mapped onto the remaining element of {a,b,c}. Finally, the fourth element of A can be mapped onto any of the three elements of {a,b,c} that are not already mapped. Therefore, the total number of onto functions in this case is:

3 × 2 × 1 × 3 = 18

Case 2: Three elements of A are mapped onto different elements of {a,b,c}, and one element is mapped onto the same element as another element

In this case, the element of A that is mapped onto the same element as another element can be chosen in 4 ways. The remaining two elements of A can be mapped onto any two different elements of {a,b,c} in 3 × 2 = 6 ways. The remaining element of {a,b,c} can be mapped onto any of the two remaining elements of A. Therefore, the total number of onto functions in this case is:

4 × 6 × 2 = 48

Case 3: Two elements of A are mapped onto the same element of {a,b,c}, and two elements are mapped onto different elements

In this case, the element of A that is mapped onto the same element as another element can be chosen in 3 ways. The remaining two elements of A that are mapped onto different elements of {a,b,c} can be chosen in 3 × 2 = 6 ways. The remaining element of {a,b,c} can be mapped onto either of the two remaining elements of A. Therefore, the total number of onto functions in this case is:

3 × 6 × 2 = 36

Case 4: All four elements of A are mapped onto the same element of {a,b,c}

In this case, there is only one onto function.

Therefore, the total number of onto functions from A to {a,b,c} is:

18 + 48 + 36 + 1 = 103

Hence, there are 103 onto functions from the set A = {1, 2, 3, 4} to the set {a, b, c}.
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How many onto functions can be defined from the set A = {1, 2, 3, 4} to {a, b, c}?
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