Exercise 2.1
Question 1
Let A={1,2,3}, B={4,5,6,7} and let f={(1,4),(2,5),(3,6)} be a function from A to B. Show that f is one-one.
Sol :
To show that a function is one-to-one (or injective), we need to demonstrate that distinct elements in the domain map to distinct elements in the codomain. In other words, if $f(a_1) = f(a_2)$, then $a_1 = a_2$.
In this case, the function f is defined as follows:
$f = \{(1,4),(2,5),(3,6)\}$
Let's check if f is one-to-one.
Assume $f(a_1) = f(a_2)$. This implies:
$ f(a_1) = f(a_2) \Rightarrow (a_1, b_1) = (a_2, b_2) $
Now, substitute the values from the given function f :
$(a_1, b_1) = (a_2, b_2) \Rightarrow (a_1, b_1) = (a_2, b_2) $
Since the first components are equal, \(a_1 = a_2\). Therefore, f is one-to-one.
In simpler terms, since each element in set A is paired with a distinct element in set B in the function f, we can conclude that f is a one-to-one function.
Question 2
Let $f: N \rightarrow N$ be defined by $f(x)=2 x$. Show that $f$ is one-one but not onto.
Question 3
Let $f: R \rightarrow R$ be defined by $f(x)=x^{2}$. Is $f$ one-to-one ?
Sol :
Question 4
Let $f: Z \rightarrow Z$ be defined by $f(x)=-x$. Show that $f$ is a bijective function.
Sol :
Question 5
Examine in each case whether the function is onto, one-one or bijective. Justify your answer.
(i) $f: R \rightarrow R$ defined by $f(x)=1+x^{2}$
(ii) $f: R \rightarrow R$ defined by $f(x)=3-4 x$
Sol :
Question 6
Let $A=\{1,2,3), B=\{4,5\}$ and let $f=\{(1,4),(2,5),(3,5)\}$. show that f is not one-one but onto function from A to B.
Sol :
Question 7
Let $f: A \rightarrow B$ be one-to-one function such that range of $f=\{b\}$. Determine the number of elements in A.
Sol :
Question 8
If N is the set of natural numbers and $A=\{0,1\}$, then prove that $f: N \rightarrow A$ defined by $f(2 x)=0$ and $f(2 x+1)=1, \forall x \in N$ is a many-one onto function.
Sol :
Question 9
(i) If $A=\{1,-1,2,-2\}$ and $B=\{2,5,7\}$, then show that $f: A \rightarrow B$ defined by $f(x)=x^{2}+1$ is neither one-one nor onto function.
(ii) Show that the function $f: N \rightarrow N$ defined by $f(x)=x^{2}, x \in N$ (set of all positive integers) is one-one but not onto.
Sol :
Question 10
Discuss the function $f:\{1,2,3\} \rightarrow\{0,3,7,13,14\}$ for one-one onto, where $f(x)=x^{2}+x+1$
Sol :
Question 11
Test the following functions for one-one and onto :
(i) $f: N \rightarrow N: f(x)=3 x+5$
(ii) $f: R \rightarrow R: f(x)=x^{2}$
(iii) $f: R \rightarrow R: f(x)=x^{2}+5$
(iv) $f: R \rightarrow R: f(x)=e^{x}$
(v) $f: C \rightarrow R: f(x+i y)=x, y \in R$.
where N= Set of natural numbers,
R= Set of real numbers,
C= Set of complex numbers.
Sol :
Question 12
Check the injectivity and surjectivity of the following functions :
(i) $f: N \rightarrow N$ defined by $f(x)=3 x$
(ii) $f: N \rightarrow N$ defined by $f(x)=x^{2}+1$
(iii) $f: Z \rightarrow Z$ defined by $f(x)=x^{2}$
(iv) $f: Z \rightarrow Z$ defined by $f(x)=2 x+1$
(v) $f: Z \rightarrow Z$ defined by $f(x)=x^{3}$
(vi) $f: R \rightarrow R$ defined by $f(x)=x^{3}$
(vii) $f: R \rightarrow R$ defined by $f(x)=[x]$
(viii) $f: R \rightarrow Z$ defined by $f(x)=[x]$
where $[x]$ denotes the greatest integer less than or equal to $x$.
Sol :
Question 13
Let $f: R \rightarrow R$ be defined by
$f(x)=\left\{\begin{array}{r}1, \text { if } x>0 \\0, \text { if } x=0 \\-1, \text { if } x<0\end{array}\right.$
Examine the function f for one-one onto.
Sol :
Question 14
Let A be the set of all 50 students of class XII in a particular school. Let $f: A \rightarrow N$ be a function defined by $f(x)=$ roll number of the student $x$. Show that $f$ is one-one but not onto.
Sol :
Question 15
Let A and B be two non-empty sets. Prove that there exists a one - one onto function from $A \times B$ to $B \times A$.
Sol :
Question 16
Show that a one-one function $f:\{1,2,3\} \rightarrow\{1,2,3\}$ must be onto.
Sol :
Question 17
Show that an onto function $f:\{1,2,3\} \rightarrow\{1,2,3\}$ is necessarily one-one.
Sol :
Question 18
Examine the type of the following function $f: R \rightarrow R$ defined by
(i) $f(x)=x^{3}$
(ii) $f(x)=\left\{\begin{array}{r}1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is irrational }\end{array}\right.$
Sol :
Question 19
Show that the function $f: R \rightarrow R$ defined by f(x)=x-[x]. where [x] denotes the integral part of x is neither one-one nor onto.
Sol :
Question 20
Show that the function $f: N \rightarrow I$ defined by
$f(x)=\left\{\begin{array}{l}\frac{1}{2}(n-1), \text { when } n \text { is odd } \\ -\frac{1}{2} n, \text { when } n \text { is even }\end{array}\right.$
is bijective
Sol :
Question 21
If A={2,3,4}, B={2,5,6,7} then construct a function which is
(i) one-one
(ii) many-one.
Sol :
Question 22
If $A=\{1,3,5\}, B=\{2,4,6\} .$ Then find all possible one-one functions from $A$ to $B$.
Sol :
Question 23
Let $A=\{1,2,3\}$. Find all one-to-one functions from $A$ to $A$.
Sol :
Question 24
Let A={-1,0,1} and $f=\left\{\left(x, x^{2}\right): x \in A\right\}$. Show that $f: A \rightarrow A$ is neither one-to-one nor onto.
Sol :
Question 25
Determine whether the function given below is one-one:
To each state of India assign its capital.
Sol :
Question 26
Let A be a finite set. If $f: A \rightarrow A$ is one-to-one. Show that f is onto.
Sol :
Question 27
Let A and B be two sets each with a finite number of elements. If there is an injective function from $A$ to $B$ and there is an injective function from $B$ to $A$. prove that there is a bijective function from $A$ to $B$.
Sol :
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