From 2e0ba3e83397e4946643121d541d9fd62e73efec Mon Sep 17 00:00:00 2001
From: zacharias <zacharias@localhost>
Date: Tue, 2 Sep 2025 16:01:37 +0200
Subject: [PATCH] sep-2-16_00

---
 exercise-sep-2.txt | 15 +++++++++++++++
 notes-sep-2.txt    | 21 +++++++++++++++++++++
 2 files changed, 36 insertions(+)
 create mode 100644 exercise-sep-2.txt
 create mode 100644 notes-sep-2.txt

diff --git a/exercise-sep-2.txt b/exercise-sep-2.txt
new file mode 100644
index 0000000..fbb8fb0
--- /dev/null
+++ b/exercise-sep-2.txt
@@ -0,0 +1,15 @@
+U = {x € Z+ : x <= 10}, A = {x : x udda, x != 5}, B = {x : x primtal}.
+
+Bestäm |A|, |B|, ¨A, ¨B, AuB, A↓uB, och A\B
+
+A = {1, 3, 7, 9}
+B = {2, 3, 5, 7}
+
+|A| = 4
+|B| = 4
+¨A = {2, 4, 5, 6, 8, 10}
+¨B = {1, 4, 6, 8, 9, 10}
+AuB = {1, 2, 3, 5, 7, 9}
+A↓uB = {3, 7}
+A\B = {1, 9}
+A"delt"B = {1, 2, 5, 9}
diff --git a/notes-sep-2.txt b/notes-sep-2.txt
new file mode 100644
index 0000000..fc839e2
--- /dev/null
+++ b/notes-sep-2.txt
@@ -0,0 +1,21 @@
+Mändoperation:
+- Differens: A\B = {x € U | x € A och x !€ B}
+- Symetrisk Differens A "delt" B = {x € U | x € AuB och x !€ A↓uB}
+- A\B != B\A
+
+Mänder 2:
+
+Räkne regler:
+- Dublekomplement: ¨¨A = A
+- Idempotens: AuA = A, A↓uA = A
+- Identitet: Auø = A, A↓uU = A
+- Dominans: AuU = U, A↓uø = ø
+- Kommutativ: AuB <=> BuA, A↓uB <=> B↓uA
+- Associativ: (AuB)uC <=> Au(BuC), (A↓uB)↓uC <=> A↓u(B↓uC)
+  - Paranteser: om alla oprationer är u (union) eller ↓u (snit) spelar årder inge roll
+  - Ex: AuBuC eller A↓uB↓uC är okej men inte AuB↓uC eftersom Au(B↓uC) != (AuB)↓uC
+- De Morgans: ¨(AuB) <=> ¨A↓u¨B, ¨(A↓uB) <=> ¨Au¨B
+  - Bevis 1:
+    x € ¨(AuB) => x !€ AuB => {x !€ A, x !€ B => x !€ {x € ¨A, x € ¨B => x € ¨(AuB). Visar att ¨(AuB) …c A↓uB.
+    x € ¨A↓u¨B => {x € ¨A, x € ¨B => {x !€ A, x !€ B => x !€ AuB => x € ¨(AuB). Visar att ¨A↓uB …c ¨(AuB).
+    Resultat: ¨(AuB) = ¨Au¨B