Question

Show that these statements about the real number x are equivalent:

(i) \(x\)is irrational,

(ii) \(3x + 2\) is irrational,

(iii) \(\frac{x}{2}\) is irrational.

Short answer

The statements mentioned in the question are equivalent.

Step-by-step solution

Introduction

Irrational numbers are the ones which cannot be written in \(\frac{p}{q}\) form where \(q \ne 0\) and \(p,q \in Z\) along with the numbers which are not-terminating and not-repeating.

Consider the statements as given:

Let,

\(P1:x\)is irrational

And,

\(P2:3x + 2\)is irrational

And,

\(P3:\frac{x}{2}\) is irrational.

To prove \(P1 \to P2\)

When x is irrational then x is a number which is not of \(\frac{p}{q}\) form where \(q \ne 0\) and\(p,q \in Z\).

Since, the multiplication of a rational and an irrational is irrational number, so 3x is an irrational number.

Also, since the addition of rational and irrational gives an irrational number.

Hence,\(3x + 2\)is irrational number.

This implies that \(P1 \to P2\)

To prove \(P2 \to P3\)

Let \(3x + 2\) is irrational.

Since, the addition of rational and irrational gives irrational number.

So\(3x + 2( - 2)\)is irrational.

Since, the multiplication of a rational and an irrational is irrational.

And since 6 is a rational, so\(\left( {3x} \right)\left( {\frac{1}{6}} \right)\)is an irrational.

Hence\(\frac{x}{2}\)is irrational.

This implies that \(P2 \to P3\)

To prove \(P3 \to P1\)

When \(\frac{x}{2}\) is irrational.

Since, the multiplication of a rational and an irrational number is irrational.

So\(2\left( {\frac{x}{2}} \right)\)is also an irrational.

Hence, \(x\) is an irrational.

This implies that\(P3 \to P1\).

Now, that implies that \(P1 \to P2 \to P3 \to P1\).Therefore, the statements mentioned in the question are equivalent.