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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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Section
BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Let O be the set of all odd integers. Prove that O has the same cardinality as 2Z, the set of all even integers.

To determine

To prove:

O has the same cardinality as 2Z.

O°οΟ

Explanation

Given information:

O be the set of all odd integers.

Z: Set of even integers.

Concept used:

A function is said to be one-to-one function if distinct elements in domain must be mapped with distinct elements in codomain.

A function is onto function if each element in codomain is mapped with at least one element in domain.

Proof:

Let set of all odd integers be Ο.

The objective is to prove that set of all odd integers has the same cardinality as the set of all even integers.

Consider that Ο is the set of all odd integers and 2Z is the set of all even integers.

The objective is to show that Ο and 2Z have same cardinality.

Define a function f:Ο2Z such that f(n)=2n for all n2Z.

Check for one-to-one.

Let n,m2Z such that,

f(n)=f(m)

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