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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Use the results of exercises 27, 31, and 32 to prove the following: If R is the set of all solutions to all equations of the from x 2 + b x + c = 0 , where b and c are integers, then R is countable.

To determine

To prove:

If R is a set of all solutions to all equations of the form x2+bx+c=0 where b and c are integers then R is countable.

Explanation

Given information:

R is a set of all solutions to all equations of the form x2+bx+c=0.

Concept used:

If A is any countably infinite set, B is any set, and g:AB is onto, then B is countable.

A union of any two countable sets is countable.

A disjoint union of any finite set and any countably infinite set is countably infinite.

A union of any two countably infinite sets is countably infinite.

Calculation:

Consider R is the set of all solutions to all equations of the form x2+bx+c=0, where b and c are integers.

The object is to prove is countable.

Use the following statements to prove is countable as follows:

1) If A is any countably infinite set, B is any set, and g:AB is onto, then B is countable.

2) Union of any two countable sets is countable.

3) The cartesian product of the set of integers with itself is countably infinite.

Let the quadratic equation be x2+bx+c=0 where b,c are integers.

The real solution of the given quadratic form is b±b24ac2a.

Let us denote these two values by αiβi which depends on b,c and a0

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