Proving a formula ∀x:x∈A∧|x−1|<a⟹x=1

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Discussion Overview

The discussion revolves around proving a formula related to a set and a condition involving a variable \( a \). Participants are examining the implications of the statement for the set \( A = \{1, 2\} \) and the nature of \( a \) as a positive real number. The focus is on the logical structure and requirements for the proof.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes the statement that for the set \( A = \{1, 2\} \), there exists an \( a > 0 \) such that the condition \( \forall x: x \in A \wedge |x - 1| < a \Longrightarrow x = 1 \) holds.
  • Another participant questions the clarity of the statement regarding whether \( A \) is indeed \( \{1, 2\} \) and seeks clarification on the origin of \( a \).
  • A later reply reiterates the uncertainty about \( A \) and specifies that \( a \) is a real number, but does not resolve the initial ambiguity.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the definition of the set \( A \) and the nature of \( a \). There is no consensus on the clarity of the initial statement or the conditions required for the proof.

Contextual Notes

The discussion highlights limitations in the clarity of the problem statement, particularly concerning the definitions of the set \( A \) and the variable \( a \). There are unresolved questions about the implications of the proposed formula.

solakis1
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Given the set {1,2},prove that there axists an a>0 such that:

$$\forall x$$:$$x\in A \wedge |x-1|<a\Longrightarrow x=1$$
 
Last edited:
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It is not clear from the statement if $A=\{1,2\}$. Also, what set does $a$ come from?
 
Evgeny.Makarov said:
It is not clear from the statement if $A=\{1,2\}$. Also, what set does $a$ come from?
[sp]Sorry, A={1,2} and a is a real No [/sp]
 
Then one can take $a=1/2$.
 

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