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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

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


Classify the following function as injection, surjection or bijection :

 f : R → R, defined by f(x) = x3 − x


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) = x and g(x) = |x| .


Consider f : {1, 2, 3} → {abc} and g : {abc} → {apple, ball, cat} defined as f (1) = af (2) = bf (3) = cg (a) = apple, g (b) = ball and g (c) =  cat. Show that fg and gof are invertible. Find f−1g−1 and gof−1and show that (gof)−1 = f 1o g−1


If A = {1, 2, 3} and B = {ab}, write the total number of functions from A to B.


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


Let f : R → R be defined as  `f (x) = (2x - 3)/4.` write fo f-1 (1) .


Let

f : R → R be given by

\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]

where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
 


(d) one-one and onto


Which of the following functions from

\[A = \left\{ x : - 1 \leq x \leq 1 \right\}\]

to itself are bijections?

 

 

 


Let  \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation

\[fog \left( x \right) = gof \left( x \right)\] is 



Mark the correct alternative in the following question:
Let f :  \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\]  R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,

 


Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1 


If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1


Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.


Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.


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

f(x) = `x/2`


If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.


Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.


Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.


The smallest integer function f(x) = [x] is ____________.


Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.


If N be the set of all-natural numbers, consider f: N → N such that f(x) = 2x, ∀ x ∈ N, then f is ____________.


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?

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


If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f is ______.


Let a function `f: N rightarrow N` be defined by

f(n) = `{:[(2n",", n = 2","  4","  6","  8","......),(n - 1",", n = 3","  7","  11","  15","......),((n + 1)/2",", n = 1","  5","  9","  13","......):}`

then f is ______.


Let A = {1, 2, 3, ..., 10} and f : A `rightarrow` A be defined as

f(k) = `{{:(k + 1, if k  "is odd"),(     k, if k  "is even"):}`.

Then the number of possible functions g : A `rightarrow` A such that gof = f is ______.


The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.


Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then

Assertion (A): f(x) has a minimum at x = 1.

Reason (R): When `d/dx (f(x)) < 0, ∀  x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀  x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.


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