Question

Translate each of the following sentences into symbolic logic. There exists a real number \(a\) for which \(a+x=x\) for every real number \(x\).

Short answer

\(\exists a \in \mathbb{R}, \forall x \in \mathbb{R}, a + x = x \)

Step-by-step solution

Identify the elements

The sentence mentions that there exists a real number \(a\), thus there is an existential quantifier. Also, it states that \(a+x=x\) for every real number \(x\), showing the presence of a universal quantifier. The task is to compose these elements into a formal mathematical statement.

Translate into symbolic logic

The sentence translated into symbolic logic will be \(\exists a \in \mathbb{R}, \forall x \in \mathbb{R}, a + x = x \). This indicates that there is at least one real number \(a\) (existential quantifier) for which the property \(a + x = x\) holds true for every real number \(x\) (universal quantifier).