Commerce Students - Class 12 Mathematics Questions

Question:

Are the following set of ordered pairs functions? If so examine whether the mapping is injective or surjective.

(i) { (x, y): x is a person, y is the mother of x }.

 (ii) { (a, b) : a is a person, b is an ancestor of a}.

Solution:

(i) The set of ordered pairs given here represents a function.

Here, the images of distinct elements of x under f are not distinct, so it is not injective but it is surjective.

(ii) Since, each element of domain does not have a unique image.

Therefore, the set of ordered pairs given here does not represent function.

Question:

Using the definition, prove that the function f : A → b is invertible if and only if f is both one-one and onto.

Solution:

A function f :X → Y is defined to be invertible, if there exist a function g = Y → X such that gof = I_x and fog = I_y. The function is called the inverse of f and is denoted by f

We know that only bijective functions are invertible functions. A bijective function is both injective and surjective.

It means function f: X → Y is invertible iff f is a bijective function.

Question:

If the set a contains 5 elements and the set b contains 6 elements, then the number of one-one and onto mappings from a to b is

(a) 720      

(b) 120      

(c) 0          

(d) None of these

Solution: (c)

Since, the number of elements in B is more than A.

Hence, there cannot be any one-one and onto mapping from A to B.

Question:

Every function is invertible.

Solution: False

We know that only bijective functions are invertible.

Question:

A binary operation on a set has always the identity element.

Solution: False

We have '+' is a binary operation on the set N but it has no identity element.

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