Question

Write the following as English sentences. Say whether they are true or false. $$ \forall X \in \mathscr{P}(\mathbb{N}), X \subseteq \mathbb{R} $$

Short answer

The mathematical sentence \(\forall X \in \mathscr{P}(\mathbb{N}), X \subseteq \mathbb{R}\) translates to 'For all sets \(X\) in the power set of natural numbers, \(X\) is a subset of the set of real numbers', and this sentence is true.

Step-by-step solution

Break Down the Mathematical Symbols

The first step is to break down the mathematical symbols into English. The \(\forall\) symbol means 'for all', \(\in\) stands for 'is an element of', \(\mathscr{P}(\mathbb{N})\) refers to the power set of natural numbers, \(X\) is a variable name, \(\subseteq\) stands for 'is a subset of', and \(\mathbb{R}\) refers to the set of real numbers.

Translate the Mathematical Sentence into English

Now, the full sentence can be translated. \(\forall X \in \mathscr{P}(\mathbb{N}), X \subseteq \mathbb{R}\) in English translates to 'For all sets \(X\) in the power set of natural numbers, \(X\) is a subset of the set of real numbers.'

Assess Truth Value of the Sentence

The last step is to assess whether the sentence is true or false. Given that the set of natural numbers is indeed a subset of the set of real numbers, it stands to reason that any subset of the natural numbers (which is what the power set of natural numbers consists of) would also be a subset of the real numbers. Therefore, the sentence is true.