Find the number of subsets $S$ of $X$ (of any size) that satisfy the following property

Question

Let $X=\{1,2,\dots,10\}$ define the relation $R$ on $X$ by:
for all $a,b\in X$, $a\mathrel{R}b \iff ab$ is even.

1) Find the number of subsets $S$ of $X$ (of any size) that satisfy the following property: $\forall a\in S,\exists b\in S,a\mathrel{R}b$. explain

I think is 5 because, $b=[2],[4],[6],[8],[10]$?

2) Find and simplify the number of two element subsets $S$ of $X$ that satisfy the following property $\forall a\in S,a\mathrel{R}1$

I don't quite get the question..
ty

Answer 1

HINTS:

1. A set $S\subseteq X$ has the property that $\forall a\in S\,\exists b\in S(a\,R\,b)$ if and only if $S$ contains an even number. (Why?) How many subsets of $X$ contain an even number? (You may find it easier to start by counting that subsets of $X$ that do not contain an even number.)

2. Note that $a\,R\,1$ if and only if $a$ is even. (Why?) How many $2$-element subsets of $X$ contain only even numbers?