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In this article, we are discussing how to find number of functions from one set to another. For understanding the basics of functions, you can refer this: Classes (Injective, surjective, Bijective) of Functions.
Nội dung chính
- How many onto functions are there from a set with 6 elements to a set with 4 elements?
- How many total functions are there from a set with three elements to a set with four elements?
- How many functions are there from a set with 3 elements to a set with 5 elements?
- How many functions are there from a set with two elements to a set with three elements?
Number of functions from one set to another: Let X and Y are two sets having m and n elements respectively. In a function from X
to Y, every element of X must be mapped to an element of Y. Therefore, each element of X has ‘n’ elements to be chosen from. Therefore, total number of functions will be n×n×n.. m times = nm.
For example: X = a, b, c and Y = 4, 5. A function from X to Y can be represented in Figure 1.
Considering all possibilities of mapping elements of X to
elements of Y, the set of functions can be represented in Table 1.
Examples: Let us discuss gate questions based on this:
- Q1. Let X, Y, Z be sets of sizes x, y and z respectively. Let W = X x Y. Let E be the set of all subsets of W. The number of functions from Z to E is:
(A) z2xy
(B) z x 2xy
(C)
z2x + y
(D) 2xyzSolution: As W = X x Y is given, number of elements in W is xy. As E is the set of all subsets of W, number of elements in E is 2xy. The number of functions from Z (set of z elements) to E (set of 2xy elements) is 2xyz. So the correct option is (D)
- Q2. Let S denote the set of all functions f: 0,14 → 0,1. Denote by N the number of
functions from S to the set 0,1. The value of Log2Log2N is ______.
(A) 12
(B) 13
(C) 15
(D) 16Solution: As given in the question, S denotes the set of all functions f: 0, 14 → 0, 1. The number of functions from 0,14 (16 elements) to 0, 1 (2 elements) are 216. Therefore, S has 216 elements. Also, given, N denotes the number of function from S(216 elements) to 0, 1(2 elements).
Therefore, N has 2216 elements. Calculating required value,Log2(Log2 (2216)) =Log216 = 16
Therefore, correct option is (D).
Number of onto functions from one set to another – In onto function from X to Y, all the elements of Y must be used. In the example of functions from X = a, b, c to Y = 4, 5, F1 and F2 given in Table 1 are not onto. In F1, element 5 of set Y is unused and element 4
is unused in function F2. So, total numbers of onto functions from X to Y are 6 (F3 to F8).
- If X has m elements and Y has 2 elements, the number of onto functions will be 2m-2.
Explanation: From a set of m elements to a set of 2 elements, the total number of functions is 2m. Out of these functions, 2 functions are not onto (If all elements are mapped to 1st element of Y or all elements are mapped to
2nd element of Y). So, number of onto functions is 2m-2. - If X has m elements and Y has n elements, the number if onto functions are,
Important notes
–
- The formula works only if m ≥ n.
- If m < n, the number of onto functions is 0 as it is not possible to use all elements of Y.
Q3. The number of onto functions (surjective functions) from set X = 1, 2, 3, 4 to set Y = a, b, c is:
(A) 36
(B) 64
(C) 81
(D) 72
Solution: Using m = 4 and n = 3, the number of onto functions is:
34 – 3C1(2)4
+ 3C214 = 36.
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Consider functions from a set with $5$ elements to a set with $3$ elements.
(a) How many functions are there?
(b) How many are one-to-one?
(c) How many are onto?
a) Each element mapped to $3$ images.
$3 cdot 3 cdot 3 cdot 3 cdot 3$
b) $0$
c) How do I
do this?
Edit: I tried doing this way.
EDIT: There can be a set of cardinality 3,1,1 or 2,2,1.
For 3,1,1: 5C3 * 2C1 * 1C1 * 3!
For 2,2,1: 5C2 * 3C2 * 1C1 * 3!
And i realized my 3! is wrong. Should be * 3 only. Why is that so?
asked Apr 28, 2022 8:47
RStyleRStyle
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5
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You correctly found that there are $3^5$ functions from a set
with five elements to a set with three elements. However, this counts functions with fewer than three elements in the range. We must exclude those functions. To do so, we can use the Inclusion-Exclusion Principle.
There are $binom31$ ways of excluding one element in the codomain from the range and $2^5$ functions from a set with five elements to the remaining two elements
in the codomain.
There are $binom32$ ways of excluding two elements in the codomain from the range and $1^5$ functions from a set with five elements to the remaining element in the codomain.
By the Inclusion-Exclusion Principle, the number of surjective (onto) functions from a set with five elements to a set with three elements is
$$3^5 – binom312^5 + binom321^5$$
answered Apr 28,
2022 9:11
N. F. TaussigN. F.
Taussig
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Hint on c)
The “onto”-function will induce a partition of its domain (as any function) and this partition (actually the fibres of the function) will – because it is onto – have exactly $3$ elements. So to be found is in the first place how many such partitions exist. A fixed partition gives room for $3times2times1=6$ functions.
So you end up with: $$6timestextnumber of partitions on 1,2,3,4,5\text
that have exactly 3text elements$$
Also have a look here (especially the counting of partitions).
A general formula for the number of onto-functions $1,dots,n\to1,dots,k$ is: $$k!S(n,k)$$where $S(n,k)$ stands for the Stirling number of the second kind.
answered
Apr 28, 2022 8:55
drhabdrhab
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1
How many onto functions are there from a set with 6 elements to a set with 4 elements?
= 6 and S(4, 3) = 6. Thus, there are 36 onto functions.
How many total functions are there from a set with three elements to a set with four elements?
The correct choice is C. The total number of injective mappings from the set containing 3 elements into the set containing 4 elements is 4P3 = 4! = 24.
How many functions are there from a set with 3 elements to a set with 5 elements?
1 Answer. Image of each element of A can be taken in 3 ways. ∴ Number of functions from A to B = 35 = 243.
How many functions are there from a set with two elements to a set with three elements?
Thus there are 32 32 32 possible functions.
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