commit 7d1d7226392c76696957f0239b446d14e580bf06
parent 6b0757d6d46526f6cc0dcc525eea6e7760b5ee80
Author: Andrew <andrewlaack1@gmail.com>
Date: Mon, 22 Apr 2024 20:23:33 -0500
Completed notes
Diffstat:
7 files changed, 67 insertions(+), 1 deletion(-)
diff --git a/CartesianProduct.md b/CartesianProduct.md
@@ -0,0 +1,12 @@
+:math310: :sets:
+# Cartesian Product
+
+Throughout textbook
+
+## Notes
+
+**Definition:** The Cartesian Product of two sets is the set of all ordered pairs a,b where a is contianed in A and b is contained in B.
+
+This set has a size of |A| * |B|.
+
+Cartesian products or cartesian sets, are denoted using x as in A x B. This is also how we describe the coordinate planes denoted as R^2 as it is the cartesian set of all real numbers and all real numbers.
diff --git a/Contradiction.md b/Contradiction.md
@@ -0,0 +1,8 @@
+:proofs: :math310:
+# Contradiction
+
+Throughout textbook
+
+## Notes
+
+**Definition:** Contradiction is used to prove if then statements. This is done by assuming the then is true and the if is not true which would imply the statement is false. From here, you then show this causes a contradiction thus if the if is true then the then is true.
diff --git a/Contrapositive.md b/Contrapositive.md
@@ -0,0 +1,8 @@
+:math310: :proofs:
+# Contrapositive
+
+Throughout TB
+
+## Notes
+
+**Definition:** To prove and if then statement with contrapositive we assume the then statement is false. Following from here we then prove the if part must also be true for the then to be false. So it follows that if the first is true then the second is also true because the second is never true when the first is false.
diff --git a/Math310.md b/Math310.md
@@ -6,4 +6,12 @@ This is the index for my main Math 310 notes.
## Main Links
[[Induction.md]]
-
+[[StrongInduction.md]]
+[[SmallestCounterExample.md]]
+[[CounterExample.md]]
+[[VectorSpace.md]]
+[[PowerSet.md]]
+[[Contrapositive.md]]
+[[Contradiction.md]]
+[[CartesianProduct.md]]
+[[ProveSetEquality.md]]
diff --git a/PowerSet.md b/PowerSet.md
@@ -0,0 +1,12 @@
+:sets: :math310:
+# Power Set
+
+AM Ch1
+
+## Notes
+
+**Definition:** The power set is the set of all subesets of the input set.
+
+Example:
+
+P(A) = {X : X $\in$ A}
diff --git a/ProveSetEquality.md b/ProveSetEquality.md
@@ -0,0 +1,8 @@
+:math310: :proofs:
+# Prove Set Equality
+
+AM TB Ch8
+
+## Notes
+
+To prove that two sets are equivalent (A and B), we first prove that A contains B. We then show that B also contains A thus all elements must be the same making the sets equivalent. Equivalence of sets is done using the = sign not the $\equiv$ sign.
diff --git a/VectorSpace.md b/VectorSpace.md
@@ -0,0 +1,10 @@
+:math310: :linear-algebra:
+# Vector Space
+
+AM Ch8
+
+## Notes
+
+**Definition:** A vector space is a set of elements that obey certain properties.
+
+An interesting thing about vector spaces is the union of two vector spaces is itself a vector space.
