How many one-to-one correspondences are there between 2 sets with 6 elements each?
The first element from the first set can be paired with any of the other 6 elements.The second element from the first set can be paired with any of the remaining 5 elements from the second set.…So, 6x5x4x3x2x1 = 6! = 720 would be the total number of combinations.
How many one-one correspondents are there between two sets with 8 elements?
Iggy Rocko and xyzzy are both correct despite each getting a thumbs down. To illustrate, let X = {1, 2, 3, 4, 5, 6, 7, 8,} and Y = {A, B, C, D, E, F, G, H} and let F be the family of one-one functions f: X → Y x ↦ y Consider one such function f. There are eight possible mappings such that f(1) = y Once this mapping is determined, there are seven possible mappings f(2) six possible mappings f(3) and so on down to one possible mapping for f(8) Given that each function is one-one, there are 8 x 7 x 6 x ...... x 2 x 1 = 8! = 40320 possible one-one functions f ∈ F Indeed, the image of each function f(1)f(2)f(3)f(4)f(4)f(5)f(6)f(7)f(8) will be a permutation of the elements of Y. For example one such function f could be represented by the string f(1)f(2)f(3)f(4)f(4)f(5)f(6)f(7)f(8) = BHADECGF Such functions could be labelled f_1, f_2, f_3, .........., f_40320 if you like. Contrast this with the number of functions not necessarily one-one nor onto. Each function f can map any element of x to any element of y. Since each element in x has independently 8 choices there are 8⁸ = 16777216 such functions f
How many one-to-one functions are there from a set with [math]m[/math] elements to a set with [math]n[/math] elements?
If [math]m>n[/math], there aren't any.If [math]m\leq n[/math], there are [math]n[/math] options for where to send the first element, [math]n-1[/math] options for the second, [math]n-2[/math] for the third and so on.So, the total number of 1:1 functions from an [math]m[/math]-set to an [math]n[/math]-set is[math]\displaystyle n(n-1)(n-2)\cdots(n-m+1)=\frac{n!}{(n-m)!}[/math]
Math correspondence questions?
1. how many one-to-one correspondences are there between the sets a b c d and 1 2 3 4 if in each correspondence if: a. b must correspond to 3 That leaves a with 3 possible correspondences. Any choice of correspondence for a will leave c with 2 possible correspondence. Any choices of correspondences for a and c will determine the 1 possible correspondence for d. 3 * 2 * 1 = 3! = 6 b. b must correspond to 3 and d to 4 Following similar reasoning to (a), 2 * 1 = 2! = 2 c. a and c must correspond to even numbers. Either a corresponds to 2 and c to 4 or vice versa (2 choices). Either b corresponds to 1 and d to 3 or vice versa (2 choices). 2 * 2 = 4 2.How many seven-digit numbers are there when 0 and 1 cannot be the leading number? There are 8 possible first digits, excluding 0 and 1. Each of the other digits has 10 possibilities, so there are 10^6 = 1,000,000 possible sequences for those digits. 8 * 1,000,000 = 8,000,000 3. A 2 digit number has the property that the units digit is 4 less than the tens digit and the tens digit is twice the units digit. What is the number? Let u represent the units' digit and t the tens' digit. u = t - 4 t = 2u u = 2u - 4 4 = u t = 2u = 8 The number is 84.
Which of the following pairs of sets can be placed in one-to-one correspondence?
"in mathematics, one-to-one correspondence refers to a situation in which the members of one set (call it A) can be evenly matched with the members of a second set (call it B)." Read more: http://science.jrank.org/pages/4861/One-One-Correspondence.html#ixzz0FIn1JASM&B so in A there are 5 numbers and 5 letters therefore there are one number for every letter. it is also the same for C where there are 13 letter and 13 numbers
What is the basic difference between a finite set and an infinite set?
I am writing here the basic difference among 3 categories..1 : Finite Set2 : Countable Infinite Set3 : Uncountable Infinite Set..We can differentiate the above 3 categories just by checking their countability. But the point is how to check the countability…In FINITE SETS , of course the elements are countable, eg: A = { 3, 5, 6, 9 } , B = { a, e, i, o, u} C = { x : x <50, x belongs to N } etc etc. In all these examples, cardinality is very clear.But in INFINITE SETS : the elements can be potentially counted or can not be counted. Like, we start with the infinite set with smallest cardinality…Set of Natural numbers N-> Infinite, countableSet of Whole numbers W-> Infinite, countableSet of Integers Z -> Infinite, countableSet of Rational numbers Q -> Infinite,countableSet of Irrational numbers I -> Infinite, uncountableSet of Real numbers R -> Infinite, uncountableAn infinite set is countable if there exists bijective mapping , ie there is one to one correspondence between the set elements & the Natural number. That means, we are able to arrange the elements of the set into simple one row, or into rows & columns. & we are very sure which number comes next. And natural number wise we can arrange the elements.. like 1st element, 2nd element, 3rd element………so on…Only thing that, these rows & columns continue to infinity…….For example…natural numbers 1,2,3,4,5,6,7,……….. infinityWhole numbers O,1,2,3,4,5,………… infinityIntegers negative infinity….. -4,-3,-2,-1,0,1,2,3,4,5 ……infinityRationals 1/1, 1/2,1/3,1/4,……..2/1, 2/2,2/3,2/4,2/5 …….3/1,3/2,3/3,3/4,………4/1,4/2,4/3,4/4,4/5 ……..If we continue this way, we can notice that, all possible fractions will be fit in the above list in one or the other rows & columns.So, all above are countable infinite sets.Now, as we know that the set of Real number is the union of two sets, Rationals & IrrationalsThe set of Real numbers are uncountable as between every 2 real numbers there is another rational & irrational numbers, So bijective mapping between the elements & the natural numbers is not possible.So, set of real numbers is uncountable & the set of Rationals are countable . So the set of irrational has to be uncountable. If it is not so the set of real numbers will become countable, which is not so…
Use finite differences and a system of equations to find a polynomial function that fits the data?
Finite differences are when you take successive differences between the f(x) values. -5...-6...-1....16....51.....110 ...-1....5....17....35....59 .......6...12....18....24 ..........6.....6......6 The degree of polynomial (i.e.the highest power) is how many rows down you have to go to get - or be free to choose - a constant difference. In this case, it's 3 (because you can fill the bottom row with 6 to build the series). See here for how it works http://www.math.hmc.edu/funfacts/ffiles/10002.3-5.shtml So that means your polynomial has the form f(x) = ax^3 + bx^2 + cx + d Now you can create a system of equations (four, because you have four unknowns) by putting x and f(x) into the polynomial: a + b + c + d = -5 ................for x=1 8·a + 4·b + 2·c + d = -6 .......for x=2 27·a + 9·b + 3·c + d = -1 ......for x=3 64·a + 16·b + 4·c + d = 16 ...for x=4 Then solve that system of simultaneous equations by substitution or whatever method you use. The put the values a,b,c,d back into f(x) = ax^3 + bx^2 + cx + d and check they generate the required f(x) for x=1-6
How many functions are there from the set X = {1,2,3,4,5} to the set Y = {a,b, c}? How many are 1-1?
Let |X| = m and |Y| = nNo. of functions f:X->Y is |Y|^|X| = n^m = 3^5 = 243.No. of onto functions is n!*S(m,n) = 3!*S(5,3) = 6*25 = 150where S(m,n) is the stirling no. of the second kind.No. of 1-1 functions is 0 (since m > n)
What are some examples of a set?
Intuitively, we can think of a set as a container that can contain anything we put in it. We customarily use curly braces to indicate the container, and put inside the curly braces the elements contained within the set. Here are some examples:The set of the first five letters of the English alphabet: {a, b, c, d, e}The set of positive integers strictly less than seven: {1, 2, 3, 4, 5, 6}The empty set: { }These are finite sets, i.e., sets that contain a finite number of elements. There are also infinite sets. Some examples of infinite sets are as follows:The set of all positive integers: {1, 2, 3, 4, 5, 6, … }The set of all even positive integers: {2, 4, 6, 8, 10, … }The set of all prime numbers: {2, 3, 5, 7, 11, 13, 17, … }These sets are called countably infinite sets because each can be put in one-to-one correspondence with the natural numbers we use for counting. There are also uncountably infinite sets, which are strictly greater in size than countably infinite sets. We cannot specify their elements by listing them. Some examples of uncountably infinite sets are as follows:The set of real numbersThe set of points on a lineThe set of points in three-dimensional spaceThe set of all subsets of the natural numbers
Can one infinite set be larger than another infinite set?
Maybe what you would like to learn about is Transfinite number theory by Georg Cantor. There indeed exists several "infinite numbers" (in fact there is an infinity of infinite, or transfinite, numbers), that is, different sizes of infinite sets. For example, the set of all real numbers is infinitely larger than the set of integers.This theory also brings some counter-intuitive results, such as (that will be in contradiction with one of the previous answer to this question) :- There are as many negative integers, than there are integers- There are as many even (respectively odd) integers, than there are integers- There are as many numbes between 0 and 1, than there are between 0 and 2- There are as many points on the surface of a square than there are on its side- etc.Just for fun, here is a little proof of the second assertion: "There are as many even (respectively odd) integers, than there are integers". The idea is to find a one-to-one correspondance between the two sets we want to compare, that is, find a correspondance between each element of the even integers set, with each element of the the integers set. This is easily done by considering that an even number can be written [math]2 \times k[/math], with [math]k[/math] any integer. So you won't find any integer that is not associated with an even integer following this methods (by absurd, if you did, that would mean that this integer [math]x[/math] is such that [math]2 \times x[/math] doesn't exist). Conclusion: they have the same number of elements.To go a bit further, one of Cantor great open problem, was to know whether or not we could find a set of infinite size between reals and integers, whose size wouldn't be the size of integers or reals (Continuum hypothesis)