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QUESTION: 1

Let A be the set of all 25 students of Class X in a school. Let f : A → N be function defined by f (x) = roll number of the student x. Then f is:

Solution:

f(x) = Roll number of student x

f(x_{1}) = Roll number of student x_{1}

f(x_{2}) = Roll number of student x_{2}

Putting f(x_{1}) = f(x_{2})

Roll number of student x_{1} = Roll number of student x_{2}

Since no two student of a class have same roll number.

Therefore, x_{1} ≠ x_{2}

So, f is one-one.

QUESTION: 2

A function f: A x B → B x A defined by f (a, b) = (b, a) on two sets A and B. The function is:

Solution:

f:A×B→B×A is defined as f(a,b)=(b,a).

Let (a1,b1),(a2,b2)∈A×B such that f(a1,b1)=f(a2,b2).

⇒(b1,a1)=(b2,a2)

⇒(b1=b2) and (a1=a2)

⇒(a1,b1)=(a2,b2)

∴f is one-one.

Now, let (b,a)∈B×A be any element.

Then, there exists (a,b)∈A×B such that

f(a,b)=(b,a). [By definition of f]

∴ f is onto.

QUESTION: 3

Let f : N→ R - {0} defined as f(x) = 1/x where x ∈ N is not an onto function. Which one of the following sets should be replaced by N such that the function f will become onto? **(where R _{0 }= R - {0})**

Solution:

QUESTION: 4

The function f is

Solution:

The function is bijective (one-to-one and onto or one-to-one correspondence) if each element of the codomain is mapped to by exactly one element of the domain. (That is, the function is both injective and surjective.) A bijective function is a bijection.

QUESTION: 5

A function f: R → R is defined by f(x) = [x+1], where [x] the greatest integer function, is:

Solution:

**Correct Answer :- b**

**Explanation :** f: R → R is given by,

f(x) = [x+1]

It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1.

∴ f(1.2) = f(1.9), but 1.2 ≠ 1.9.

∴ f is not one-one.

Now, consider 0.7 ∈ R.

It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7.

∴ f is not onto.

Hence, the greatest integer function is neither one-one nor onto.

QUESTION: 6

How many onto functions from set A to set A can be formed for the set A = {1, 2, 3, 4, 5, ……n}?

Solution:

Taking set {1,2,3}

Since f is onto,all elements of{1,2,3} have unique pre-image.

Total no. of one-one function = 3×2×1=6

Eg:- Since f is onto,all elements of {1,2,3} have unique pre-image.

total no. of onto functions = n×n−1×n−2×2×1

= n!

QUESTION: 7

Let A = {1, 2, 3}. f: A → A Then complete the function f such that it one-one and onto: f = {(1, 2), (2,1) ______}

Solution:

QUESTION: 8

If a relation f: X → Y is a function, then for g: Y → X to be a function, function f need to be

Solution:

QUESTION: 9

For which one of these mapping the function f(x) = x^{2} will be one-one?

Solution:

consider ‘f’ is a function whose domain is set A. The function is said to be injective if for all x and y in A,

Whenever f(x)=f(y), then x=y

And equivalently, if x ≠ y, then f(x) ≠ f(y)

Formally, it is stated as, if f(x) = f(y) implies x=y, then f is one-to-one mapped, or f is 1-1.

QUESTION: 10

Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = 3x + 4. Then the composition of f and g is ____________

Solution:

The composition of f and g is given by f (g(x)) which is equal to 2 (3x + 4) + 1.

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