# Thread: For a set, how many different relations are there?

1. ## For a set, how many different relations are there?

There are a lot of parts to this, which probably aren't that difficult for most. But I'd like to see how accurate my work is (up to this point). Some of my answers are shown, though I doubt I'm right for all of them (if any).

Let $A$ be the set $\{h,o,m,e,1,3,t,r,i,b\}$. Answer the following:

a) How many different relations can be defined on the set $A$?
My current answer is $2^{n^2}=2^{10^2}=2^{100}$

b) How many different relations $T$ are there on $A$ where $(1,3)\in T$ and $(t,r)\in T$?
My current answer is $2^{n^2-2}=2^{10^2-2}=2^{98}$, although I'm very doubtful the answer is right.

c) How many have $(m,e)\ni T$? (I've used this symbol for "not in" or "not an element", since Latex doesn't have a strike-through option)
My current guess is $2^{n^2}-2^{n^2-1}=2^{n^2-1}=2^{99}$, but this is probably wrong. I really have no idea.

d) How many relations on $A$ are there that are not functions?
My current answer is $2^{n^2}-n^n=2^{10^2}-10^{10}$

e) How many different reflexive functions are there?
My current answer is $2^{n^2-n}=2^{10^2-10}=2^{90}$

f) How many different symmetric relations are there?
My current answer is $2^{C(n+1,2)}=2^{n(n+1)/2}=2^{10(11)/2}=2^{55}$, though I'm really unsure about this one.

Can anyone help with this?

2. Very well done. All are correct.
BTW: $$\notin$$ gives $\notin$.