Are most real functions non-linear?

Question

I made an observation that for two finite sets $A$, $B$ that most $R \subseteq A \times B$ where $R$ is a function also 'appear to be' non-linear. It got me wondering if this is true in the highly general case of functions from $\mathbb{R}^n$ to $\mathbb{R}^m$.

Let $\Omega$ be the set of a linear functions $f: \mathbb{R}^n \mapsto \mathbb{R}^m$ and $\Xi$ be the set of all non-linear maps $g: \mathbb{R}^n \mapsto \mathbb{R}^m$.

Is there some measure $\mu : Q \mapsto \mathbb{R}_{\geq 0}$ (or other precise way of quantifying the "size" of sets) that shows whether $\mu(\Omega) \leq \mu (\Xi)$?

Answer 1

The set of linear functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ is in correspondence with the set of $m\times n$ matrices with real coefficients. This is a basic result of Linear Algebra. This set has the same cardinality as the reals, namely $\mathfrak{c}=2^{\aleph_0}$.

The set of all functions from $\mathbb{R}$ to itself has cardinality $|\mathbb{R}|^{|\mathbb{R}|} = 2^{\mathfrak{c}}$, which by Cantor's Theorem is strictly larger than $\mathfrak{c}=|\mathbb{R}|$. The same is true for the set of all functions from $\mathbb{R}^n$ to $\mathbb{R}^m$.

Answer 2

The other answers provide a nice cardinality-centric answer to the question. As an alternative, I don't believe it's particularly hard to show that the set of linear functions is a closed meager subset of the topological space of continuous functions $C(\mathbb{R}^n,\mathbb{R}^m)$ with the compact-open topology.