Solution:
Functions are the backbone of advanced mathematics topics like calculus. Functions are of many types, like into and onto. Let's solve a problem regarding onto functions.
To find the number of onto functions from set A [with m elements] and set B [with n elements], we have to consider two cases:
⇒ One in which m ≥ n: In this case, the number of onto functions from A to B is given by:
→ Number of onto functions = nm - nC1[n - 1]m + nC2[n - 2]m - ....... or as [summation from k = 0 to k = n of { [-1]k . nCk . [n - k]m }].
Let's solve an example.
→ Let m = 4 and n = 3; then using the above formula, we get 34 - 3C1[3 - 1]4 + 3C2[3 - 2]4 = 81 - 48 + 3 = 36.
Hence, they have 36 onto functions.
⇒ One in which m < n: In this case, there are no onto functions from set A to set B, since all the elements will not be covered in the range function; but onto functions from set B to set A is possible in this case though.
Hence, The formula to find the number of onto functions from set A with m elements to set B with n elements is nm - nC1[n - 1]m + nC2[n - 2]m - ....... or [summation from k = 0 to k = n of { [-1]k . nCk . [n - k]m }], when m ≥ n.
Write the formula to find the number of onto functions from set A to set B.
Summary:
The formula to find the number of onto functions from set A with m elements to set B with n elements is nm - nC1[n - 1]m + nC2[n - 2]m - ... or [summation from k = 0 to k = n of { [-1]k . nCk . [n - k]m }], when m ≥ n.
[a] | [i] How many functions are there from a set with three elements to a set with four elements? Explain your answer. | [2] |
[ii] How many functions are there from a set with four elements to a set with three elements? Explain your answer. | [2] | |
[i] The 1st element can map to one of the elements in 2nd group. The 2nd element can also map to one of the elements in 2nd group [as the 2nd element can have same range as the 1st element]. Same for the 3rd element. \ no. of functions = 4 * 4 * 4 = 64 [ii] The 1st element have a selection of 3 elements in 2nd group, so as the 2nd, 3rd, and 4th elements. Nội dung chính Show
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[b] | [i] How many one-to-one functions are there from a set with three elements to a set with four elements? Explain your answer. | [2] |
[ii] How many one-to-one functions are there from a set with four elements to a set with three elements? Explain your answer. | [2] | |
[i] The 1st element has a selection of 4 elements. The 2nd element has a selection of 3 elements. The 3rd element has a selection of 2 elements. \ no. of one-to-one functions = 4 * 3 * 2 = 24
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[c] | [i] How many onto functions are there from a set with three elements to a set with four elements? Explain your answer. | [2] |
[ii] How many onto functions are there from a set with four elements to a set with three elements? Explain your answer. | [2] | |
[i] No onto functions as the number of elements in the first group is less than the numbers of elements in the 2nd group. [ii] [3 * 1 * 2 * 1] + [3 * 2 * 2 * 1] + [3 * 2 * 1* 3] = 6 + 12 + 18 = 36 | ||
[d] | [i] How many invertible functions are there from a set with three elements to a set with four elements? | [1] |
[ii] How many invertible functions are there from a set with four elements to a set with three elements? | [1] | |
[i] No invertible functions. In order to have invertible functions, it must be one-to-one and onto. In this case, since the number of elements in the 1st group is less the 2nd group. It is no onto function and therefore no invertible functions. [ii] No invertible functions. As it has no one-to-one function [refer to b[ii]], therefore no invertible functions. | ||
[e] | Construct a function from three elements to two elements which is neither one-to-one nor onto. | [2] |
All three elements in the domain must map onto one element in the range. | ||
[f] | The relation R, "is a factor of " is defined on . Decide whether or not R is reflexive, symmetric or transitive. | [4] |
R is reflexive as every number is a factor of itself. R is not symmetric as 3 is a factor of 6 but 6 is not a factor of 3. R is transitive as if a is a factor of b, and b is a factor of c, then a must also be a factor of c. |
How many onto functions are there from a set with 4 elements to a set with 3 elements?
An onto function from a set of 4 elements to a set of 3 elements must map two of the four elements to one of the three elements. There are C[4, 2] C[3,1]=18 ways to do this.
How many one to one functions are there from a set with five elements to sets with 5 elements?
[d] 2520 one-to-one functions.
How many one to one functions are there from a set with five elements to sets with four elements?
The objective is to find the number of one-to-one functions is there from a set with 5 elements to set with 4 elements. Here so there are no one-to-one functions from the set with 5 elements to the set with 4 elements. Therefore, there are one-to-one functions from the set with 5 elements to the set with 4 elements.
How many one to one functions are there from a set containing 5 elements to a set containing 7 elements?
How many functions are there from a 5-element set to a 7-element? this element, so the total number of possible assignments is 7 · 7 · 7 · 7 · 7=75 . Thus, [c] is the correct answer.