Prove that the cardinality of the set of all reflexive relations over $\Bbb N$ is $\mathfrak c$

Question

How to prove that the cardinality of the set of all reflexive relations over $\Bbb N$ is $\mathfrak c$?

I know that

1. The cardinal number of $|\mathcal P(\Bbb N\times \Bbb N)| = \mathfrak c$.
2. The set $R$ of all reflexive relations over $\Bbb N$ is a subset of $\mathcal P(\Bbb N\times \Bbb N)$ (in other words: $R \subseteq \mathcal P(\Bbb N\times \Bbb N)$ )
3. Because of (2), $|R|\leq\mathfrak c$

I need to prove that $R$ is also $|R|\geq\mathfrak c$, and then I can use Cantor–Schröder–Bernstein theorm to prove $ |R| = \mathfrak c $

Please advise. I'm new to set theory and cardinals, so please explain it clearly.

Thank you in advance!

Answer 1

A relation $R\subseteq \mathbb{N}\times\mathbb{N}$ is reflective if and only if $R$ contains the diagonal $\Delta = \{(n,n) : n\in\mathbb{N}\}$. Therefore each subset $A\subseteq \mathbb{N\times N} \setminus \Delta$ gives the reflective relation.

We must check the powerset $\mathcal{P}(\mathbb{N\times N} \setminus \Delta)$ has cardinality $\mathfrak{c}$. This is because the set $\mathbb{N\times N} \setminus \Delta$ has cardinality $\aleph_0$ and the cardinality of power set of a countable is the cardinality of continuum.