Proving a formula ∀x:x∈A∧|x−1|<a⟹x=1

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In summary, the symbol ∀ stands for "for all" and is used in mathematical logic to indicate that a formula applies to all elements. The expression x∈A means that x is an element of the set A. The significance of |x−1|<a in the formula is that it represents a condition that must be true for the formula to hold. This formula uses logical deduction to prove that if x∈A and |x−1|<a, then x=1. The practical application of this formula is in mathematical and scientific contexts to prove the value of x when certain conditions are met.
  • #1
solakis1
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Given the set {1,2},prove that there axists an a>0 such that:

\(\displaystyle \forall x\):\(\displaystyle x\in A \wedge |x-1|<a\Longrightarrow x=1\)
 
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  • #2
It is not clear from the statement if $A=\{1,2\}$. Also, what set does $a$ come from?
 
  • #3
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]
 
  • #4
Then one can take $a=1/2$.
 

Related to Proving a formula ∀x:x∈A∧|x−1|<a⟹x=1

1. What is the meaning of the symbol ∀ in the formula?

The symbol ∀ stands for "for all" and is part of the universal quantifier notation in mathematical logic.

2. What does x∈A mean in the formula?

The expression x∈A means that x is an element of the set A, indicating that the formula applies to all elements in the set.

3. What is the significance of |x−1|

This expression represents a condition or constraint that must be true for the formula to hold. In this case, it means that the absolute value of the difference between x and 1 must be less than a.

4. How does this formula prove that x=1?

This formula uses logical deduction to prove that if the conditions x∈A and |x−1|

5. What is the practical application of this formula?

This formula can be used to prove the value of x in situations where the given conditions are met. For example, it could be used in mathematical proofs or in scientific experiments to show that a certain value must equal 1 based on specific conditions.

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