Question

Translate each of the following sentences into symbolic logic. For every positive number \(\varepsilon,\) there is a positive number \(\delta\) for which \(|x-a|<\delta\) implies \(|f(x)-f(a)|<\varepsilon\)

Short answer

The symbolic logic translation of the sentence is: \(\forall \varepsilon > 0, \exists \delta > 0, 0 < |x-a| < \delta \Rightarrow |f(x)-f(a)|<\varepsilon\)

Step-by-step solution

Recognize

Recognize the concept in the sentence : 'For every positive number \(\varepsilon\), there is a positive number \(\delta\) for which \(|x-a|<\delta\) implies \(|f(x)-f(a)|<\varepsilon\).' This sentence is the formal definition of the limit in calculus, often referred to as the \(\varepsilon\)-\(\delta\) definition of the limit.

Identify variables and constants

Identify the variables and constants in the statement. Here, \(x\) and \(a\) are variables, and \(f(x)\) and \(f(a)\) are constants.

Translate into Symbolic Logic

Translate each component of the sentence into its corresponding symbolic logic. The sentence translates to: 'For all \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x-a| < \delta\), then \(|f(x)-f(a)|<\varepsilon\).' In symbolic logic, this becomes \(\forall \varepsilon > 0, \exists \delta > 0, 0 < |x-a| < \delta \Rightarrow |f(x)-f(a)|<\varepsilon\).