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Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______ - Mathematics

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प्रश्न

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______

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उत्तर

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is n!

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अध्याय 1: Relations And Functions - Solved Examples [पृष्ठ १०]

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एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12
अध्याय 1 Relations And Functions
Solved Examples | Q 27 | पृष्ठ १०

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Show that the function f : R* → R* defined by f(x) = `1/x` is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true if the domain R* is replaced by N, with the co-domain being the same as R?


Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x2


Let fR → be defined as f(x) = 10x + 7. Find the function gR → R such that g o f = f o = 1R.


Give an example of a function which is neither one-one nor onto ?


Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3


Classify the following function as injection, surjection or bijection :

f : Q → Q, defined by f(x) = x3 + 1


Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4


Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = 1 + x2


Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.


Find gof and fog when f : R → R and g : R → R is  defined by  f(x) = 8x3 and  g(x) = x1/3.


 Find fog and gof  if  : f (x) = ex g(x) = loge x .


Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.


Let fgh be real functions given by f(x) = sin xg (x) = 2x and h (x) = cos x. Prove that fog = go (fh).


State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}


If f : Q → Qg : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.


 If f : R → R be defined by f(x) = x4, write f−1 (1).

If f : R → R is defined by f(x) = x2, find f−1 (−25).


If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).


Let f be an injective map with domain {xyz} and range {1, 2, 3}, such that exactly one of the following statements is correct and the remaining are false.

\[f\left( x \right) = 1, f\left( y \right) \neq 1, f\left( z \right) \neq 2 .\]

The value of

\[f^{- 1} \left( 1 \right)\] is 

 


If the function\[f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}\] is a surjection, then A =

 

 


\[f : Z \to Z\]  be given by

 ` f (x) = {(x/2, ", if  x is even" ) ,(0 , ", if  x  is  odd "):}`

Then,  f is


If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\] 

 


Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.


Which function is used to check whether a character is alphanumeric or not?


If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))


Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}


Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

f = {(1, 4), (1, 5), (2, 4), (3, 5)}


Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

k(x) = x2 


The number of bijective functions from set A to itself when A contains 106 elements is ____________.


Which of the following functions from Z into Z is bijective?


Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.


Given a function If as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then


An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions.

  • Ravi wants to know among those relations, how many functions can be formed from B to G?

An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions.

  • Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Answer the following questions using the above information.

  • Let : N → R be defined by f(x) = x2. Range of the function among the following is ____________.

Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:


If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` is ______.


The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.


Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.



The given function f : R → R is not ‘onto’ function. Give reason.


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