Relations: Reflexive, symmetric, transitive I am having difficulties determining if this relation is reflexive, symmetric, transitive, or none of these.
Let A be the set of all strings of $0's$, $1's$, and $2's$ of length $4$. Define a relation $R$ on $A$ as follows: For all $s$, $t ∈ A$, $s$ R $t$ if and only if the sum of the characters in s equals the sum of the characters in t.
Examples this is true:
$0121$ R $2200$ 
$1220$ R $2111$
 A: Reflexive means $aRa$ for every $a$. But clearly sums are the same for $a$ and $a$, so it is reflexive.
Symmetric means if $aRb$ then $bRa$ - so since $aRb$, both $a,b$ have the same sum, so $bRa$ is true.
Transitive means $aRb$ and $bRc$ imply $aRc$. Note that $aRb$ and $bRc$ means $a,b$ and $b,c$ have same sum, so $a,c$ must have it too, and thus $R$ is transitive.
Any relation with all 3 properties is called an equivalence relation, and this one is an equivalence relation indeed.
A: Tip: Any relation that is defined like $aRb$ iff $f(a)=f(b)$ for some function $f$ is an equivalence. Try if you can prove that.
Protip: The converse also holds by passing to the quotient.
A: HINT: I suspect that you’re letting yourself think it harder than it really is.


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*Reflexivity: If $a_1a_2a_3a_4\in A$, is it true that $a_1+a_2+a_3+a_4=a_1+a_2+a_3+a_4$? That’s exactly what’s required in order for $R$ to be reflexive.

*Symmetry: If $a_1a_2a_3a_4,b_1b_2b_3b_4\in A$ and $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4$, is it true that $b_1+b_2+b_3+b_4=a_1+a_2+a_3+a_4$? That’s exactly what’s required in order for $R$ to be symmetric.
Now see if you can deal with transitivity completely on your own.
