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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