Advertisements
Advertisements
Question
Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
Advertisements
Solution
Formula:
When two sets A and B have m and n elements respectively, then the number of onto functions from A to B is
\[\binom{ \sum\nolimits_{r = 1}^n \left( - 1 \right)^r n C_r r^m , \text{if m} \geq n}{\text{o if m} < n }\]
Here, number of elements in A = 4 = m
Number of elements in B = 2 = n
So, m > n
Number of onto functions
\[= \sum\nolimits_{r = 1}^2 \left( - 1 \right)^r 2 C_r r^4 \]
\[ = \left( - 1 \right)^1 2 C_1 1^4 + \left( - 1 \right)^2 2 C_2 2^4 \]
\[ = - 2 + 16\]
= 14
APPEARS IN
RELATED QUESTIONS
Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Show that the function f : R → R given by f(x) = x3 is injective.
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 3), (b, 2), (c, 1)}
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 2), (b, 1), (c, 1)}
Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and f, g : A → B be functions defined by f(x) = x2 – x, x ∈ A and g(x) = `2|x - 1/2| – 1`, x ∈ A. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + x2
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Let R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Find fog and gof if : f (x) = ex g(x) = loge x .
Find fog and gof if : f (x) = |x|, g (x) = sin x .
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)} [NCERT EXEMPLAR]
Let
\[f : R \to R\] be a function defined by
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
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.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.
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}
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Which of the following functions from Z into Z is bijective?
Let g(x) = x2 – 4x – 5, then ____________.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
The function f : R → R given by f(x) = x3 – 1 is ____________.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.
Let f: R → R defined by f(x) = x4. Choose the correct answer
Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
Many students choosing the same favourite subject resembles which type of function?
