Hashing: Fixed a couple of typos in notation

06-hash/hash.tex

@@ -198,7 +198,7 @@ Now, we turn back to the original linear family~$\cal L$.
 \theorem{The family~${\cal L}$ is $(2,4)$-independent.}
 
 \proof
-We fix $x,y\in [p]$ distinct and $i,j\in [p]$. To verify $(2,4)$-independence,
+We fix $x,y\in [p]$ distinct and $i,j\in [m]$. To verify $(2,4)$-independence,
 we need to prove that $\Pr[h_{a,b}(x) = i \land h_{a,b}(y) = j] \le 4/m^2$ for a~random
 choice of parameters $(a,b)$.
 
@@ -251,7 +251,7 @@ For 2-independence, we proceed in a~similar way. We are given two items $x_1\ne
 and two buckets $j_1,j_2\in [m]$. We are bounding
 $$
 	\Prsub{h}[h(x_1) \bmod m = j_1 \land h(x_2) \bmod m = j_2] =
-	\sum_{i_1\equiv j_1\atop i_2\equiv j_2} \Pr[h(x_1) = i_1 \land h(h_2) = i_2].
+	\sum_{i_1\equiv j_1\atop i_2\equiv j_2} \Pr[h(x_1) = i_1 \land h(x_2) = i_2].
 $$
 Again, each term of the sum is at most $c/r^2$. There are at most $\lceil r/m\rceil \le (r+m-1)/m$
 choices of~$i_1$ and independently also of~$i_2$. The sum is therefore bounded by
@@ -936,7 +936,7 @@ This filter can be analysed similarly to the $k$-band version. We will assume th
 all hash functions are perfectly random and mutually independent.
 
 Insertion of $n$~elements sets $kn$ bits (not necessarily distinct), so the
-probability that a~fixed bit $B[i]$ is set is $(1-1/m)^{nk}$, which is approximately
+probability that a~fixed bit $B[i]$ is unset is $(1-1/m)^{nk}$, which is approximately
 $p = \e^{-nk/m}$. We will find the optimum value of~$p$, for which the probability
 of false positives is minimized. For fixed~$m$, we get $k = -m/n\cdot\ln p$.