diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 5dcede8..5ddbfb9 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -36,9 +36,9 @@
               "type": "pdf",
               "state": {
                 "file": "Concurrent Systems/slides/class 6.pdf",
-                "page": 2,
+                "page": 3,
                 "left": -24,
-                "top": 282,
+                "top": 72,
                 "zoom": 0.6448931116389548
               },
               "icon": "lucide-file-text",
@@ -220,6 +220,8 @@
   },
   "active": "51157f32453cba69",
   "lastOpenFiles": [
+    "Pasted image 20250318090909.png",
+    "Pasted image 20250318090733.png",
     "Concurrent Systems/slides/class 6.pdf",
     "Concurrent Systems/notes/6 - Atomicity.md",
     "Concurrent Systems/slides/class 5.pdf",
@@ -255,7 +257,6 @@
     "Concurrent Systems/notes/images/Pasted image 20250304082459.png",
     "Concurrent Systems/notes/images/Pasted image 20250304093223.png",
     "Concurrent Systems/notes/images/Pasted image 20250304090219.png",
-    "Concurrent Systems/notes/images/Pasted image 20250304090207.png",
     "Foundation of data science/notes/1 CV Basics.md",
     "Foundation of data science/notes/7 Autoencoders.md",
     "Foundation of data science/notes/6 PCA.md",
diff --git a/Concurrent Systems/notes/6 - Atomicity.md b/Concurrent Systems/notes/6 - Atomicity.md
index c7c8b66..e08a37e 100644
--- a/Concurrent Systems/notes/6 - Atomicity.md	
+++ b/Concurrent Systems/notes/6 - Atomicity.md	
@@ -14,9 +14,12 @@ A history is **complete** if every inv is eventually followed by a corresponding
 ### Linearizability
 A complete history $\hat{H}$ is **linearizable** if there exists a sequential history $\hat{S}$ s.t.
 - $\forall X :\hat{S}|_{X} \in semantics(X)$
-- $\forall p:\hat{H|_{p} = \hat{S}|p}$
+- $\forall p:\hat{H}|_{p} = \hat{S}|p$
 - If $res[op] <_{H} inv[op']$, then $res[op] <_{S} inv[op']$
 	- can rearrange events only if they overlap
 
+Given an history $\hat{K}$, we can define a binary relation on events $⟶_{K}$ s.t. (op, op’) ∈ ⟶K if and only if res[op] <K inv[op’]. We write op ⟶K op’ for denoting (op, op’) ∈ ⟶K. Hence, condition 3 of the previous Def. requires that ⟶H ⊆ ⟶S.
 
+![[Pasted image 20250318090733.png]]
 
+![[Pasted image 20250318090909.png]]
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