I am new here and this is the first time that I am asking a question here, and it's also the first time that I am using $\LaTeX$, so I apologize if I make some mistakes. I study in a non-English university, so I am sorry if my questions are a bit flimsy. Due to universities being closed, I have to learn everything from home with some class notes that my teacher put online. I am taking a discrete mathematics class for the first time, and I find it do-able, but it gets a bit hard to study it by myself. I have been doing some exercises regarding relations, but I ran into some difficulties for these two questions:
1) Let $R$ be a relation defined on $\mathbb{R}_+ \times \mathbb{R}_+$ where $(x_1,y_1)\mathrel{R}(x_2,y_2)$ if and only if $x_1 \times y_2 \leq x_2 \times y_1$. Prove or refute the following : $R$ is a partial order.
2) Let $R_1$ and $R_2$ be two relations defined on $X$. Show that if $R_1$ and $R_2$ are equivalence relations, then $R_1 \cap R_2$ defined on $X$ is also a equivalent relation.
Here is what I have been able to do so far:
1) I don't know how to do this. I really need help with this kind of question.
2) To show that $R_1 \cap R_2$ is an equivalence relation when $R_1$ and $R_2$ are equivalence relations, I supposed that if $x\mathrel{(R_1 \cap R_2)}y$ and $y\mathrel{(R_1 \cap R_2)}z$, then $x \mathrel{R_1} y$ and $y \mathrel{R_1} z$, so $x \mathrel{R_1} z$ and for $R_2$ we have $x \mathrel{(R_1 \cap R_2)} z$. I think that this proves that $R_1 \cap R_2$ is transitive, but I am not sure. Assuming this is correct (please explain how), how could I phrase this in a proper way?
Thanks! Your help is really appreciated.