Problem 78

We will analyze each proposition to determine its validity.

1. Proposition ${p}\_{1}$: If a complex number $z$ satisfies $\\dfrac{1}{z}∈R$, then $z∈R$.

Let $z=a+bi$ where $a,b∈R$. If $\\dfrac{1}{z} = \\dfrac{1}{a+bi} = \\dfrac{a−bi}{{a}^{2}+{b}^{2}} ∈R$, then $b=0$, which implies $z∈R$. Therefore, proposition ${p}\_{1}$ is true.

2. Proposition ${p}\_{2}$: If a complex number satisfies ${z}^{2}∈R$, then $z∈R$.

Let $z=a+bi$ where $a,b∈R$. If ${z}^{2} = a^{2} - b^{2} + 2abi ∈R$, then $ab=0$, which implies either $a=0$ or $b=0$. This means that $z$ could be a real number or a purely imaginary number. Therefore, proposition ${p}\_{2}$ is false.

3. Proposition ${p}\_{3}$: If complex numbers ${z}\_{1},{z}\_{2}$ satisfy ${z}\_{1}{z}\_{2}∈R$, then ${z}\_{1}= \\bar{{z}\_{2}}$.

Let ${z}\_{1}=a+bi$ and ${z}\_{2}=c+di$ where $a,b,c,d∈R$. If ${z}\_{1}{z}\_{2} = (ac-bd)+(ad+bc)i ∈R$, then $ad+bc=0$. However, this does not necessarily imply that ${z}\_{1}= \\bar{{z}\_{2}}$. Therefore, proposition ${p}\_{3}$ is false.

4. Proposition ${p}\_{4}$: If a complex number $z∈R$, then $\\bar{z}∈R$.

Let $z=a+bi$ where $a,b∈R$. If $z∈R$, then $b=0$, which implies that $z=a$. Consequently, $\\bar{z}=a∈R$. Therefore, proposition ${p}\_{4}$ is true.

The correct answer is B: $\boxed{{p}\_{1},{p}\_{4}}$.