Question

Decide if the following statements are true or false. Explain. $$ \mathbb{R}^{3} \subseteq \mathbb{R}^{3} $$

Short answer

The statement \(\mathbb{R}^{3} \subseteq \mathbb{R}^{3}\) is true because every element of \(\mathbb{R}^{3}\) is also an element of \(\mathbb{R}^{3}\). In other words, \(\mathbb{R}^{3}\) is a subset of itself.

Step-by-step solution

Understanding the symbol \(\subseteq\)

First, it's important to understand that the symbol \(\subseteq\) stands for 'is a subset of'. So, the expression \(\mathbb{R}^{3} \subseteq \(\mathbb{R}^{3}\)'' states that \(\mathbb{R}^{3}\) is a subset of \(\mathbb{R}^{3}\). This means that every element of \(\mathbb{R}^{3}\) should also be an element of \(\mathbb{R}^{3}\).

Evaluating the statement

Next, evaluate whether \(\mathbb{R}^{3}\) can be a subset of \(\mathbb{R}^{3}\). In this case, since \(\mathbb{R}^{3}\) and \(\mathbb{R}^{3}\) are the same set, every element in \(\mathbb{R}^{3}\) is also in \(\mathbb{R}^{3}\). Therefore, it's clear that \(\mathbb{R}^{3}\) is a subset of \(\mathbb{R}^{3}\).

Conclusion

The original statement, \(\mathbb{R}^{3} \subseteq \(\mathbb{R}^{3}\)'' is therefore true, as each element from \(\mathbb{R}^{3}\) is indeed present in \(\mathbb{R}^{3}\). As a result, \(\mathbb{R}^{3}\) is proven to be a subset of itself.