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By definition, a permutation is a bijection from a set to itself. Since a permutation is bijective, it is one-to-one.
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Is it true that every function is a permutation if and only if it is one to one?
no
Every finite group is isomorphic to a permutation group?
yes form cayleys theorem . every group is isomorphic to groups of permutation and finite groups are not an exception.
Every permutation can be written as a product of transposition. How to prove or just apparently?
Prove it using deduction._______First you prove, that every permutation is a product of non-intercepting cycles, which are a prduct of transpsitions
Give an example of a permutation and combination?
If there is a group of 3 coloured balls, then any groups of 2 balls selected from it will be considered as a combination, whereas the different arrangements of every combination will be considered as a permutation
Is the relation a function?
Not every relation is a function. But every function is a relation. Function is just a part of relation.
Is it true that every function is a permutation if and only if it is one to one?
no
Every finite group is isomorphic to a permutation group?
yes form cayleys theorem . every group is isomorphic to groups of permutation and finite groups are not an exception.
Every permutation can be written as a product of transposition. How to prove or just apparently?
Prove it using deduction._______First you prove, that every permutation is a product of non-intercepting cycles, which are a prduct of transpsitions
What are permutation primes?
Permutation primes are primes that have digits that can be rearranged with every possible rearrangement being a prime. Some examples of permutation prime sets are {13, 31}, {17, 71}, {79, 97}, {113, 131, 311}, and {337, 373, 733}.
Give an example of a permutation and combination?
If there is a group of 3 coloured balls, then any groups of 2 balls selected from it will be considered as a combination, whereas the different arrangements of every combination will be considered as a permutation
What is the difference between permutation and combinations?
A permutation is an arrangement of objects in some specific order. Permutations are regarded as ordered elements. A selection in which order is not important is called a combination. Combinations are regarded as sets. For example, if there is a group of 3 different colored balls, then any group of 2 balls selected from it will be considered as a combination, whereas the different arrangements of every combination will be considered as a permutation.
Is the relation a function?
Not every relation is a function. But every function is a relation. Function is just a part of relation.
How do you prove Cayley's theorem which states that every group is isomorphic to a permutation group?
Cayley's theorem:Let (G,$) be a group. For each g Є G, let Jg be a permutation of G such thatJg(x) = g$xJ, then, is a function from g to Jg, J: g --> Jg and is an isomorphism from (G,$) onto a permutation group on G.Proof:We already know, from another established theorem that I'm not going to prove here, that an element invertible for an associative composition is cancellable for that composition, therefore Jg is a permutation of G. Given another permutation, Jh = Jg, then h = h$x = Jh(x) = Jg(x) = g$x = g, meaning J is injective. Now for the fun part!For every x Є G, a composition of two permutations is as follows:(Jg ○ Jh)(x) = Jg(Jh(x)) = Jg(h$x) = g$(h$x) = (g$h)$x = Jg$h(x)Therefore Jg ○ Jh = Jg$h(x) for all g, h Є GThat means that the set Ђ = {Jg: g Є G} is a stable subset of the permutation subset of G, written as ЖG, and J is an isomorphism from G onto Ђ. Consequently, Ђ is a group and therefore is a permutation group on G.Q.E.D.
Is function always return a value?
no, every function can not return a value. for example void name() { cout<<"Hello world"; } this function does not return any value due to the key word void that tells the compiler that the function does not returns a value.
Is all relation a function?
Not every relation is a function. A function is type of relation in which every element of its domain maps to only one element in the range. However, every function is a relation.
Is every function a realation?
Yes, but not the other way round - not every relation is a function.
A function will always be a vertical line?
A function can never be a vertical line, because it then fails the definition of a function: every x value outputs only 1 y value. The vertical line test will determine if a relation is a function. If a vertical line intersects the graph of the function at more than one place, it is not a function.
