Question

is \[\mathbb{R}\] a subset of \[\mathbb{R}^2\]?

Answer 1

For \(A\subseteq B\) it must be that for all \(a \in A\) we have \(a\in B\). For starters, let's take some element of the real numbers, say \(5\in\mathbb R\). Is that element contained in \(\mathbb R^2\)?

Answer 2

yes?

Answer 3

The answer is no. Recall the definition of \(\mathbb R^2\).
\[\mathbb R^2 \equiv \mathbb R \times \mathbb R := \left\{(a,b): a,b\in\mathbb R\right\}\]In other words, it's the set of all ordered pairs of real numbers. 5, nor any element of \(\mathbb R\), is not a real ordered pair so it is not contained in \(\mathbb R^2\).

Answer 4

Thus, do we have \(\mathbb R \subseteq \mathbb R^2\) or \(\mathbb R \not\subseteq\ \mathbb R^2\)?

Answer 5

i think i see, so an element in \[\mathbb{R}^2\] must be an ordered pair?

Answer 6

By definition, yes. Think of it as the points in a plane, while \(\mathbb R\) are the points on a number line.

Answer 7

Ok I think I understand now, R is not a subset of R^2 right?

Answer 8

Exactly.

Answer 9

Thanks for the help