Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) There exists a set \(X\) for which \(\mathbb{R} \subseteq X\) and \(\varnothing \in X\).

Short answer

The statement 'There exists a set \(X\) for which \(\mathbb{R} \subseteq X\) and \(\varnothing \in X\)' is true. An example satisfying these conditions is \(X = \mathbb{R} \cup \{\varnothing\}\), where \(\mathbb{R}\) is the set of all real numbers, and \(\varnothing\) stands for the empty set.

Step-by-step solution

Understanding the Keywords

In this exercise, there are few keywords which needs a clear understanding: `subset` and `element`. When we say \( \mathbb{R} \subseteq X \), it means every element present in the set of real numbers \( \mathbb{R} \) will also be present in the set \(X\). And, when it is mentioned \(\varnothing \in X \), it implies \(\varnothing\) is an element of set \(X\).

Understanding the Difference Between Subset and Element

While it might seem the same, there is a key difference between a subset and an element of a set. \(\mathbb{R} \subseteq X\) means that \(X\) contains all the elements of \(\mathbb{R}\), but \(\mathbb{R}\) itself is not considered an element of \(X\), just the individual real numbers in it. Meanwhile, \(\varnothing \in X\) directly means that the empty set (\(\varnothing\)), treated as a single entity, is contained within \(X\).

Finding a Proof

We can create a set \(X\) which consists of both real numbers and the empty set to refute the contradiction that might arise with \(\mathbb{R} \subseteq X\) and \(\varnothing \in X\). A set is defined by its elements and let's define \(X\) in such a way that we include all real numbers and the empty set. So, \(X\) can be defined as \(X = \mathbb{R} \cup \{\varnothing\}\). This implies that the set of all real numbers is a subset of \(X\) and the empty set is an element of \(X\). Hence, this fits our requirement.