Question

Decide if the following statements are true or false. Explain. $$ \mathbb{R}^{2} \subseteq \mathbb{R}^{3} $$

Short answer

The statement is true. It is because every point in \( \mathbb{R}^{2} \) can be represented in \( \mathbb{R}^{3} \) by adding a third coordinate, providing a geometric view that \( \mathbb{R}^{2} \) is a plane within the three-dimensional space \( \mathbb{R}^{3} \). Hence, \( \mathbb{R}^{2} \) is a subset of \( \mathbb{R}^{3} \).

Step-by-step solution

Understand the notations

The notation \( \mathbb{R}^{2} \) represents the set of all ordered pairs of real numbers, which can be graphically represented as a two-dimensional plane. On the other hand, \( \mathbb{R}^{3} \) represents the set of all ordered triples of real numbers, which can be graphically depicted as a three-dimensional space.

Understand the concept of subset

A subset, defined strictly, is a set whose elements are all members of another set. This means if set A is a subset of set B, then all elements of set A are also elements of set B.

Evaluate the statement

We can view \( \mathbb{R}^{2} \) as included in \( \mathbb{R}^{3} \) by adding a third coordinate, 0, to every pair in \( \mathbb{R}^{2} \). This would place \( \mathbb{R}^{2} \) as a plane within \( \mathbb{R}^{3} \). Therefore, every point in \( \mathbb{R}^{2} \) can be found in \( \mathbb{R}^{3} \) and \( \mathbb{R}^{2} \) can be seen as a subset of \( \mathbb{R}^{3} \).