In how many ways can an onto function be defined from a domain D = {a, b, c, d} to a range, R = {u, v, w}?Correct answer is '36'. Can you explain this answer?

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In how many ways can an onto function be defined from a domain D = {a, b, c, d} to a range, R = {u, v, w}?

Correct answer is '36'. Can you explain this answer?

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In how many ways can an onto function be defined from a domain D = {a,...

An onto function is one in which all the elements of the range have at least one mapping in the domain i.e. for every y in the range there exists at least one x such that f{x) = y.
This is similar to a case where we have to arrange four different balls (a, b, c, d) into three different boxes (u, v, w), such that no box is empty. Now, the only possible way of dividing the four balls into three groups is 1, 1 and 2 balls per box.
The number of ways in which 2 balls can be selected from 4 balls = ⁴C₂ = 6 (So we have a group of 2 balls and 2 other balls)
Number of ways of placing these three different groups into three boxes = 3! = 6
Hence, the total number of ways = 6 * 6 = 36

Answer: 36

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In how many ways can an onto function be defined from a domain D = {a,...

Introduction:
An onto function, also known as a surjective function, is a function where every element in the range is mapped to by at least one element in the domain. In this case, we need to determine the number of ways an onto function can be defined from a domain D = {a, b, c, d} to a range R = {u, v, w}.

Approach:
To find the number of ways an onto function can be defined, we need to consider the possible mappings from the domain to the range. We can break down the problem into smaller cases and calculate the number of ways for each case.

Case 1: All elements in the range are mapped to:
In this case, each element in the range must be mapped to by at least one element in the domain. There are 3 elements in the range, and for each element, there are 4 possible choices from the domain. Therefore, the number of ways for this case is 4 * 4 * 4 = 64.

Case 2: One element in the range is not mapped to:
In this case, one element in the range is not mapped to by any element in the domain. There are 3 elements in the range, and we need to choose one element to be excluded from the mapping. There are 3 ways to choose the excluded element. For the remaining 2 elements in the range, there are 4 possible choices from the domain. Therefore, the number of ways for this case is 3 * 4 * 4 = 48.

Case 3: Two elements in the range are not mapped to:
In this case, two elements in the range are not mapped to by any element in the domain. There are 3 elements in the range, and we need to choose two elements to be excluded from the mapping. There are 3 ways to choose the first excluded element and 2 ways to choose the second excluded element. For the remaining element in the range, there are 4 possible choices from the domain. Therefore, the number of ways for this case is 3 * 2 * 4 = 24.

Total number of ways:
To find the total number of ways, we sum up the number of ways for each case:
64 + 48 + 24 = 136.

However, we need to consider that the function must be onto, meaning that all elements in the range must be mapped to. Therefore, we need to subtract the cases where one or two elements are not mapped to:
136 - 48 - 24 = 64.

Conclusion:
There are 64 ways to define an onto function from the domain D = {a, b, c, d} to the range R = {u, v, w}.

Answered by Wizius Careers · View profile

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