Question $1$: Let $S = \{1,2,3,4,...,9\}$. Let $R$ be the relation on the power set of $S$ defined by: for every $X,Y$ in the power set of $S$, $XRY$ iff $\{1,2,3\}$ is a subset of $X$ or $Y$.
Is the following true?
$1$: for every $X$ in the power set, there exists a set $Y$ in the power set such that $XRY$. I think this one is true because we can choose $Y$ to be $\{1,2,3\}$.
$2$: There exists a set $X$ in the power set such that for every $Y$ in the power set, $XRY$. I think this one is also true because $X$ can be $\{1,2,3\}$ and the relation will hold.
$3$: how many sets $X$ in the power set are there such that $XR\{1\}$ are there? I think the answer is $2^7$ because $X$ has to have $\{1,2\}$ in this case but for the rest of the elements we can either put them or not, it doesn't matter.
Question 2: let $A = \{1,2,3,....,250\}$. Define $R$ on $A$ cross $A$ by for any $(a,b)$ , $(c,d)$ in $A$ cross $A$, $(a,b)R(c,d)$ iff $a+b = c+d$.
1: How many equivalence classes does R have? Explain I'm not sure about this one but I think it is (250 choose 2) ? if that is true how can I explain it?
Thanks!
Agree with 1 and 2. 3 is on the right track. If by "is a subset of $X$ or $Y$" you mean "is a subset of $X\cup Y$" then the answer is right, though you meant $X$ has to have $2$ and $3$ (not $1$ and $2$). If it means "is a subset of $X$ or a subset of $Y$", then it's wrong. You need to have $\{1,2,3\}\subseteq X$ since its not a subset of $\{1\},$ so there are $2^6$ possibilities.
For the second one there are as many equivalence classes as unique sums of two elements. You can make any number from $2$ to $500.$