English

Mark the Correct Alternative in the Following Question: Let a = {1, 2, ... , N} and B = {A, B}. Then the Number of Subjections from a into B is (A) Np2 (B) 2n − 2 (C) 2n − 1 (D) Nc2 - Mathematics

Advertisements
Advertisements

Question

Mark the correct alternative in the following question:
Let A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is

Options

  •  nP2 

  • 2n - 2

  • 2n - 1

  •  nC2

MCQ
Advertisements

Solution

As, the number of surjections from A to B is equal to the number of functions from A to B minus the number of functions from A to B whose images are proper subsets of B.
And, the number of functions from a set with n number of elements into a set with m number of elements = mn
So, the number of subjections from A into B where A = {1, 2, ... , n} and B = {ab} is 2n - 2 (As, two functions can be many-one into functions)
Hence, the correct alternative is option (b).

shaalaa.com
  Is there an error in this question or solution?
Chapter 2: Functions - Exercise 2.6 [Page 79]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 2 Functions
Exercise 2.6 | Q 52 | Page 79

RELATED QUESTIONS

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x3


Let f : R → R be defined as f(x) = x4. Choose the correct answer.


Give examples of two functions fN → Z and gZ → Z such that g o f is injective but gis not injective.

(Hint: Consider f(x) = x and g(x) =|x|)


Give an example of a function which is one-one but not onto ?


Classify the following function as injection, surjection or bijection :

 f : Z → Z, defined by f(x) = x − 5 


Classify the following function as injection, surjection or bijection :

 f : R → R, defined by f(x) = sinx


Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x3 + 1


Classify the following function as injection, surjection or bijection :

f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`


Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`


Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x2 + 8 and g(x) = 3x3 + 1 .


Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2xg(x) = 1/x and h(x) = ex.


Consider f : N → Ng : N → N and h : N → R defined as f(x) = 2xg(y) = 3y + 4 and h(z) = sin z for all xyz ∈ N. Show that ho (gof) = (hogof.


If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.


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


Find fog and gof  if : f (x) = x2 g(x) = cos x .


If f(x) = |x|, prove that fof = f.


Let f : R `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f -1.


If f : A → Ag : A → A are two bijections, then prove that fog is an injection ?


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


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


Let fg : R → R be defined by f(x) = 2x + l and g(x) = x2−2 for all x

∈ R, respectively. Then, find gof.  [NCERT EXEMPLAR]


The function f : R → R defined by

`f (x) = 2^x + 2^(|x|)` is 

 


Which of the following functions form Z to itself are bijections?

 

 

 
 

Let

\[f : R - \left\{ n \right\} \to R\]

\[f\left( x \right) = \frac{x - m}{x - n}, \text{where} \ m \neq n .\] Then,
 

\[f : R \to R\] is defined by

\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]

 


Which of the following functions from

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

 


The inverse of the function

\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by

\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is 

 


Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto


Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto


The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.


The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers is ____________.


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.

  • The function f: Z → Z defined by f(x) = x2 is ____________.

Let f: R → R defined by f(x) = 3x. Choose the correct answer


A function f: x → y is/are called onto (or surjective) if x under f.


Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.


The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.


Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.


Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.

(Note P(x, y) is lattice point if x, y ∈ I)

(where [.] denotes greatest integer function)


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×