Question

Write the following as English sentences. Say whether they are true or false. $$ \forall n \in \mathbb{Z} . \exists m \in \mathbb{Z} \cdot m=n+5 $$

Short answer

The English translation of the mathematical statement is: 'For every integer \(n\), there exists an integer \(m\) such that \(m\) equals \(n + 5\)' and the truth value of this statement is true.

Step-by-step solution

Translation of Symbols

The symbol '∀' stands for 'for all' and '∃' indicates 'there exists'. The '∈' symbol means 'in' or 'belongs to' and 'ℤ' represents the set of all integers. Therefore, the given expression can be translated to 'for all n that belong to the set of integers, there exists an m that also belongs to the set of integers such that m equals n + 5'. Now we need to evaluate this sentence for truthfulness.

Assessing Truthfulness of the Statement

From the sentence we obtained during the translation, it can be determined whether it is true or false. The sentence tells that for any integer \(n\), there exists another integer \(m\) which is 5 more than \(n\). From what we know about numbers, this statement is true because for any integer, if we add 5 to it, we get another integer.