Problems & Solutions

Problem

$f:\mathbb{N}\to \mathbb{N}$ is a function defined by

$f(x)=\frac{x+1}{2}$ , if $x$ is odd

$f(x)=\frac{x}{2}$ , if $x$ is even

(a) Is the function one-one? Why/why not?

(b) Is the function onto? Why/why not?

Problem

A function $f:\mathbb{R}-\left\{ \frac{2}{3} \right\}\to \mathbb{R}-\left\{ \frac{4}{3} \right\}$ is defined by $f(x)=\frac{4x+5}{3x-2}$ .

(a) Is this function bijective? Why/why not?

(b) Find the inverse of the function if it exists.

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Problem

$f:\mathbb{N}\to \mathbb{R}-\left\{ 0 \right\}$ is a function defined by $f(x)=\frac{1}{x}$ .

(a) Is $f$ one-one? Why/why not?

(b) Is $f$ onto? Why/why not?

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Problem

Find the implicit domain & range of the function $f$ given by $f(x)=\frac{2x-3}{3x+2}$ .

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Problem

If $f:X\to Y$ is an invertible function with inverse ${{f}^{-1}}$ , show that the inverse of ${{f}^{-1}}$ is $f$ .

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Problem

Show that the function $f:\left[ -\frac{1}{2} \right.,\left. \infty \right)\to \left[ -6,\left. \infty \right) \right.$ given by $f(x)=4{{x}^{2}}+4x-5$ is invertible. Find the inverse of $f$ .

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Problem

$f:\mathbb{R}\to \mathbb{R}$ is a function defined by $f(x)={{x}^{2}}$ .

(a) Is $f$ one-one? Why/why not?

(b) Is $f$ onto? Why/why not?

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Problem

Find sets A, B & C such that $A\cap B$ , $B\cap C$ & $C\cap A$ are non-empty sets & $A\cap B\cap C=\varnothing$ .

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Problem

Let A & B be two sets. If $A\cap X=B\cap X=\varnothing$ & $A\cup X=B\cup X$ for a given set X, show that $A=B$ .

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Problem

Show that $A\cap B=A\cap C$ need not imply $B=C$ .

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Problem

A single complete graph needs to be constructed by taking apart the edges of three separate complete graphs having 3, 6 & 18 vertices respectively and assembling the edges together. Is this possible and if so, what is the order of the resultant graph?

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Problem

Given $A\subset B$ , show that $C-B\subset C-A$ .

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Problem

If A, B & C are sets such that $A\cup B=A\cup C$ and $A\cup B=A\cup C$ , show that $B=C$ .

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Problem

When the units and tens digit of a 3-digit multiple of ten are interchanged, the number obtained becomes lesser than the original number by one-fortieth of the original number. What is the original number?

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Problem

A circular pathway of uniform width needs to be built inside a circular garden such that the pathway touches the boundary of the garden throughout. Given that the garden has a radius of 100 m and the area of the pathway is 19% of the area of the area of the garden, find the width of the pathway. Also, find the ratio of the inner circumference of the pathway to its outer circumference.

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