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

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SUMMARY

The discussion centers on proving the existence of integers y and z for any positive integer x. The contrapositive approach is highlighted, stating that if x is not a positive integer, then neither y nor z can exist. The definitions of positive integers and integers are clarified, establishing the foundational concepts necessary for the proof. Participants seek guidance on how to proceed with the proof effectively.

PREREQUISITES
  • Understanding of positive integers and integers
  • Familiarity with contrapositive reasoning in mathematical proofs
  • Basic knowledge of mathematical notation and terminology
  • Experience with proof techniques in number theory
NEXT STEPS
  • Study the principles of mathematical induction for proving existence statements
  • Explore the concept of contrapositive proofs in greater depth
  • Review examples of proofs involving integers and positive integers
  • Investigate the role of existential quantifiers in mathematical logic
USEFUL FOR

Students in mathematics, particularly those studying number theory or proof techniques, as well as educators seeking to enhance their understanding of integer properties and proof strategies.

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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.


Homework Equations





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

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