diff --git a/cs3223/cheatsheet.pdf b/cs3223/cheatsheet.pdf
index 668ae8f..f9bc8e6 100644
Binary files a/cs3223/cheatsheet.pdf and b/cs3223/cheatsheet.pdf differ
diff --git a/cs3223/cheatsheet.typ b/cs3223/cheatsheet.typ
index ea1d79b..e3f2728 100644
--- a/cs3223/cheatsheet.typ
+++ b/cs3223/cheatsheet.typ
@@ -166,6 +166,10 @@
   $pi_L (R)$, [project column by list $L$ from $R$],
   $pi_L^* (R)$, [project with duplicates],
 
+  $b_d$, [Data records that can fit on page],
+  $b_i$, [Data entries that can fit on page],
+  $b_r$, [RIDs that can fit on page],
+
 
 )
 == External Merge Sort
@@ -233,9 +237,12 @@
 - *Conjunct*: $1>=$ terms connected by $or$
 - *CNF predicate*: $1>=$ conjuncts connected by $and$
 - *Covered Conjunct* - predicate $p_i$ is covered conjunct if each attribute in $p_i$ is in key $K$ or include column of Index $I$
-  - $sigma_p (R), p = ("age" > 5) and ("height" = 180) and ("level" = 3), I_1 "key" = ("level", "weight", "height"$
-  - $p_c = 
-- *Primary Conjunct* - 
+  - $p = ("age" > 5) and ("height" = 180) and ("level" = 3)$
+  - $I_1 "key" = ("level", "weight", "height")$
+  - $p_c "wrt" I_1 = ("height" = 180) and ("level" = 3)$
+- *Primary Conjunct*
+  - $I$ matches $p$ if attributes in $p$ form prefix of $K$ and all comparison operators are equality except last
+  - $p_p$ is largest subset of conjuncts in $p$ such that $I$ matches $p_p$
 - $sigma_p (R)$: Select rows from $R$ that satisfy predicate $p$
 - Access Path: way of accessing data records / entries
     - *Table Scan*: Scan all data pages (Cost: $|R|$)
@@ -248,3 +255,44 @@
 == B+ Trees
 - For Index Scan + RID Lookup, many matching RIDs could refer to same page
   - Sort matching RIDs before performing lookup: Avoid retrieving same page
+
+=== Analysis
+#{
+let nin = [$N_"internal"$]
+let nle = [$N_"leaf"$]
+let nlo = [$N_"lookup"$]
+let nso = [$N_"sort"$]
+let nco = [$N_"combine"$]
+[
+Cost of index scan = $nin + #nle + nlo$
+- *#nin*: No of internal nodes accessed
+  - Height of B+ tree index
+  - $ "height(est)" = cases(
+  ceil(log_F (ceil( (||R||) / b_d))) &"if index is clustered", 
+  ceil(log_F (ceil( (||R||) / b_i))) &"otherwise", 
+) $
+- *#nlo*: Data pages accessed for RID lookups
+  - If $I$ is covering index for $sigma_p (R), nlo = 0$
+  - else $nlo = ||sigma_p_c (R)||$
+  - If matching RIDs are sorted before RID lookup
+    - $nlo = nso + min{||sigma_p_c (R)||, |R|}$
+- *#nso*: sorting matching RIDs 
+  - $nso = 0 "if" ceil( (||sigma_p_c (R)||) / b_r ) <= B$ (if RIDs can fit into $B$)
+  - $ nso = 2 ceil((||sigma_p_c (R)||) / b_r) ceil(log_(B-1) (N_0)), N_0 = ceil(ceil((||sigma_p_c (R)||) / b_r) / B) $
+    - Sorting with External Merge Sort
+    - #nso does'nt include read IO for pass 0 as its included in #nin and #nle
+    - #nso does'nt incldue write IO for final merging pass as RID is used for lookup
+- *#nle*: Leaf pages scanned for evaluating $sigma_p (R)$
+  - $nle = ceil((||sigma_p_p (R)||)/b_d)$ if clustered
+  - $nle = ceil((||sigma_p_p (R)||)/b_d)$ if unclustered
+- *Index Combination*
+- Cost = $nin^p + nle^p + nin^q + nle^q + nco + nlo$
+  - #nco: IO cost to compute join of $pi_p$ $pi_q$
+  - If $min{|pi_X_p (S_p)|, |pi_X_q (S_q)|} <= B$
+    - One of the join operands can fit in mem, then $nco = 0$
+== Hash based Index Scan
+- Cost: $N_"dir" + N_"bucket" + nlo$
+  - $N_"dir"$: no of directory pages accessed (1 if extensible hash, 0 otherwise)
+  - $N_"bucket"$: max no of index's primary/overflow pages accessed
+  - $nlo= nso + min{||sigma_p_c (R)||, |R|}$ if I is not covering index for $sigma_p (R)$
+]}