Number of correct statements among the following. Statement I: If f , g : R R are defined as f ( x ) = { x , if x is rational 0 , if x is irrational and g ( x ) = { 0 , if x is rational x , if x is irrational , then f - g is one-one

Question

$Number of correct statements among the following.$

$Statement I: If f, g : R \to R are defined as$ $f (x) =$ ${\begin{matrix} x, & if x is rational \\ 0, & if x is irrational \end{matrix}$ $and$ $g (x) =$ ${\begin{matrix} 0, & if x is rational \\ x, & if x is irrational \end{matrix},$

$then f - g is one-one and onto.$

$Statement II: f (x) = \sin x + \cos a x is a periodic function, then a must be rational.$

$Statement III: If f : A \to B and g : B \to C are functions such that g \circ f : A \to C is one-one, then f must be one-one.$

Smart Achievers · Accepted Answer

(4)

$f - g =$ $x$ $x is rational$

$= - x$ $x is irrational$

Solution

1 1

2 2

3 0

4 3

Ans.
(4)

$f - g =$ $x$ $x is rational$

$= - x$ $x is irrational$

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Solution

1 1

2 2

3 0

4 3

Ans. (4) f-g=x x is rational =-x x is irrational

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Ans.
(4)

$f - g =$ $x$ $x is rational$

$= - x$ $x is irrational$