# Proving Existence of Integer y and z for x in Positive Integers

• chocolatelover
In summary, the conversation discusses proving the existence of an integer x in the positive integers, with corresponding integers y and z. The individual asking for help mentions using the contrapositive and clarifies the definitions of positive integers and integers. They then request a hint or further guidance.
chocolatelover

## Homework Statement

Prove that for all x there exists and x if it is an element of the positive integers, then there is an integer y and an integer z.

## The Attempt at a Solution

I know that the contrapositive would be "If there is not an element of the positive integers, then there is not an integer y or an integer z." I also know that positive integers=1,2,3,...and integers=...-2, -1, 0, 1, 2,...

Could someone please give me a hint or show me what to do from here?

Thank you very much

Can you rephrase that in a way that makes any sense whatsoever?

## 1. How do you define a positive integer?

A positive integer is any whole number greater than 0. It can be written without a decimal or fractional component.

## 2. Why is it important to prove the existence of integer y and z for x in positive integers?

Proving the existence of integer y and z for x in positive integers is important because it helps to establish the fundamental properties of positive integers and their relationships with other mathematical concepts. It also allows us to make accurate and precise calculations and predictions based on these properties.

## 3. What methods can be used to prove the existence of integer y and z for x in positive integers?

There are various methods that can be used to prove the existence of integer y and z for x in positive integers, such as mathematical induction, direct proof, and proof by contradiction. Each method involves using logical reasoning and mathematical principles to demonstrate the existence of the desired integers.

## 4. Are there any counterexamples to this statement?

No, there are no counterexamples to this statement. It is a universally accepted truth in mathematics that for any positive integer x, there exists at least one integer y and z that satisfy the given condition.

## 5. Can this statement be extended to include negative integers?

No, this statement specifically refers to positive integers and does not apply to negative integers. The existence of integers y and z for any positive integer x cannot be extended to include negative integers as the properties and relationships of negative integers are different from those of positive integers.

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