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Into Function, Bijective function, Equality of function - (Concept)
Into Function:
A function $f: X \rightarrow Y$ is said to be an into function if there exists an element in $Y$ having no preimage in A .
In other words, if $\mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y}$ is not onto mapping then it is an into mapping.
Eg
As the element y2 in codomain does not have a pre-image in domain, so it is into function
Note: If a function is not onto, then it is into and
If a function is not into, then it is onto.
Bijective Function
A function $f: X \rightarrow Y$ is said to be bijective, if $f$ is both one-one and onto (meaning it is both injective and surjective)
Consider, $X_1=\{1,2,3\}$ and $X_2=\{x, y, z\}$
Eg
$\mathrm{f}: \mathrm{x}_1 \rightarrow \mathrm{x}_2$
The number of bijective function:
If $f(x)$ is bijective, and the function is from a finite set $A$ to a finite set $B$, then
$
n(A)=n(B)=m(\text { Say })
$
And, the number of Bijective functions $=\mathrm{m}$ !
Equal Functions
The two functions $f$ and $g$ are said to be equal if
$\operatorname{Domain}(\mathrm{f})=$ Domain(g)
Co-domain(f) = Co-domain(g), and
$f(x)=g(x)$ for all $x$ belonging to domain
| Exam | Chapter |
| JEE MAIN | Sets, Relations and Functions |
Let be a function defined as
where
Show that is invertible and its inverse is?
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| D. |
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Let
Statement - 1: The set
Statement - 2: is a bijection.
Statement - 1 (Assertion) and Statement - 2 (Reason).
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If n(A) = 3 and n(B) = 5. How many bijective functions are possible from A to B?
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Which of the following function is an into function in the co-domain of all real numbers?
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Which of the following is an into function in ?
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Which of the two functions are equal?
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Which function is equal to the function $f(x)=ln\: x\; ?$
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If then the no. of elements in the set
and
is not one-one is............
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Which of the following is the mapping for into function?
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If set A has 5 elements and set B has 3 elements. How many bijective function can be formed from A to B ?
| A. |
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If n(A) = 4 and n(B) = 4 . Then how many bijective functions can be formed from A to B?
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Which of the functions are not equal?
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Let a function be defined by
then,
is
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Let be functions defined by
where
is the maximum of the powers of those primes
such that
divides
and
for all
Then, the function
is
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| B. |
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| C. |
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| D. |
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Let . Then the number of bijective functions
, such that
is equal to:
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| D. |
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Let be defined as
Then which of the following statements is true ?
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The function $f: \mathrm{N} \rightarrow \mathrm{N}$ is defined by $
f(x)=x-5\left[\frac{x}{5}\right]
$ where $\mathrm{N}$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to $x$, is:
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Which of the following function is a bijective function?
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Which of the following function are bijections?
| A. |
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| D. |
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Which of the following functions are bijections?
| A. |
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| D. |
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Find the no. of bijective functions from $X$ to $Y$, where $n(X)=5=n(Y)$.
| A. |
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Which of the following is a bijective function for $f: \mathbb R\rightarrow \mathbb R$ ?
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Which of the following is an Into function for $f: (-\infty, \infty)\rightarrow [0,\infty)$
| A. |
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Let $A=\{1,2,3,4,5,6,7\}$. The number of bijective functions $f: A \rightarrow A$ such that $f(1)=3$ \& $f(i) \neq i$ for all $i=1,2, \ldots, 7$ is
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Statement 1: $\log x^2$ and $(2 \log x)$ are equal functions
Statement 2: $\frac{x^2-4}{x-2}$ and $(x+2)$ are equal functions
Statement $3: \operatorname{sgn}(x)$ and $\frac{|x|}{x}$ are not equal functions
Which statements are correct?
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Which of the following functions from Z to Z are bijections?
(a) $f(x)=x^3$
(b) $f(x)=x+2$
(c) $f(x)=2 x+1$
(d) $f(x)=x^2+1$
| A. |
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| D. |
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Definition 7 A function f : X Y is said to be one-one and onto (or bijective), if f is
both one-one and onto.
