Determine whether the following relations are reflexive, symmetric, antisymmetric, and/or transitive:
- The empty relation $\text{R}=\{\}$ is defined on the natural numbers.
- The complete relation $\mathrm{R}=\mathbf{N} \times \mathbf{N}$ defined on the natural numbers.
- The relation $\mathrm{R}$ on the positive integers where $a \text{R} b$ means a $\mid b$.
- The relation $\mathrm{R}$ on $\{w, x, y, z\}$ where $\mathrm{R}=\{(w, w),(w, x),(x, w),(x, x),(x, z),(y, y),(z$, $y),(z, z)\}$
- The relation $\mathrm{R}$ on the integers where $a\text{R}b$ means $a^2=b^2$.
