ece108: add proof techniques

docs/1b/ece108.md

@@ -196,3 +196,60 @@ $$\forall x\in\mathbb R,\forall y\in\mathbb R\equiv \forall x,y\in\mathbb R$$
     
     $\exists y\in\mathbb R,\forall x\in\mathbb R, x-y=1$ is **false** as when $x$ is selected first, it is impossible for every value of $y$ to satisfy the open sentence.
 
+## Proof techniques
+
+There are a variety of methods to prove or disprove statements.
+
+- **Deduction**: a chain of logical inferences from a starting assumption to a conclusion
+- **Case analysis**: exhausting all possible cases (e.g., truth table)
+- **Contradiction**: assuming the conclusion is false, which follows that a core assumption is false, therefore the conclusion must be true
+- **Contrapositive**: is equivalent to the original statement
+- **Counterexample**: disproves things
+- **Induction**: Prove for a small case, then prove that that applies for all cases
+
+Implications can be proven in two simple steps:
+
+1. It is assumed that the hypothesis is true (the implication is always true when it is false)
+2. Proving that it follows that the conclusion is true
+
+!!! example "Proving implications"
+    Prove that if $n+7$ is even, $n+2$ is odd.
+    
+    $\text{Proof:}$
+    
+    $\text{Assume }n+7\text{ is an even number. It follows that for some }k\in\mathbb Z$
+    
+    $$
+    \begin{align*}
+    n+7&=2k \\
+    \text{s.t.} n+2&=2k-5 \\
+    &=2(k-3)+1
+    \end{align*}
+    $$
+    
+    $\text{which is of the form }2z+1,z\in\mathbb Z,\text{ thus } n+2\text{ is odd.}$
+
+!!! example "Proof by contradiction"
+    Prove that there is no greatest integer.
+    
+    $\text{Proof:}$
+    
+    $\text{ Let }n\in\mathbb Z\text{ be given and assume }\overbrace{\text{for the sake of contradiction}^\text{FTSOC}}\text{ that }n\text{ is the largest integer. Note that }n+1\in\mathbb Z\text{ and }n+1>n.\text{ This contradicts the initial assumption that }n\text{ is the largest integer, therefore there is no largest integer.}$
+
+### Formal theorems
+
+An **even number** is a multiple of two.
+
+$$\boxed{n\ \text{is even}\iff\exists k\in\mathbb Z,n=2k}$$
+
+An **odd number** is a multiple of two plus one.
+
+$$\boxed{n\text{ is odd}\iff\exists k\in\mathbb Z,n=2k+1}$$
+
+A number is **divisible** by another if it can be part of its product.
+
+$$\boxed{n\text{ is divisible by } m\iff\exists k\in\mathbb Z,n=mk}$$
+
+A number is a **perfect square** if it is the square of an integer.
+
+$$n\text{ is a perfect square}\iff \exists k\in\mathbb Z,n=k^2$$