ece108: add laws

docs/1b/ece108.md

@@ -119,3 +119,80 @@ $$p\implies q\text{ is the converse of }q\implies p$$
 A **contrapositive** is the negatated converse, and is **logically equivalent to the original implication**. This allows proof by contrapositive.
 
 $$\neg p\implies\neg q\text{ is the contrapositive of }q\implies p$$
+
+### Operator laws
+
+Both **AND** and **OR** are commutative.
+
+$$
+p\wedge q\equiv q\wedge p \\
+p\vee q\equiv q\vee p
+$$
+
+Both **AND** and **OR** are associative.
+
+$$
+(p\wedge q)\wedge r\equiv p\wedge(q\wedge r) \\
+(p\vee q)\vee r\equiv p\vee(q\vee r)
+$$
+
+Both **AND** and **OR** are distributive with one another.
+
+$$
+p\wedge(q\vee r)\equiv(p\wedge q)\vee(p\wedge r) \\
+p\vee(q\wedge r)\equiv(p\vee q)\wedge(p\vee r)
+$$
+
+!!! tip "Proof"
+    $$
+    \begin{align*}
+    (\neg p\vee\neg r)\wedge s\wedge\neg t&\equiv\neg(p\wedge r\vee s\implies t) \\
+    \tag*{definition of implication} &\equiv \neg (p\wedge r\vee[\neg s\vee t]) \\
+    \tag*{DML} &\equiv\neg(p\wedge r)\wedge\neg[(\neg s)\vee t)] \\
+    \tag*{DML} &\equiv(\neg p\vee\neg r)\wedge\neg[(\neg t)\vee t] \\
+    \tag*{DML} &\equiv(\neg p\vee\neg r)\wedge\neg(\neg s)\wedge\neg t \\
+    \tag*{double negation} &\equiv(\neg p\vee\neg r)\wedge s\wedge\neg t
+    \end{align*}
+    $$
+
+### Quantifiers
+
+A **quantified statement** includes a **quantifier**, **variable**, **domain**, and **open sentence**.
+
+$$
+\underbrace{\text{for all}}_\text{quantifier}\  \underbrace{\text{real numbers}\overbrace{x}^\text{variable}\geq 5}_\text{domain}, \underbrace{x^2-x\geq 0}_\text{open sentence}
+$$
+
+The **universal quantifier** $\forall$ indicates "for all".
+
+$$\forall x\in S,P(x)$$
+
+!!! example
+    All real numbers greater than or equal to 5, defined as $x$, satisfy the condition $x^2-x\geq 0$.
+    
+    $$\forall x\in\mathbb R\geq 5,x^2-x\geq 0$$
+
+The **existential quantifier** $\exists$ indicates "there exists at least one".
+
+$$\exists x\in S, P(x)$$
+
+!!! example
+    There exists at least one real number greater than or equal to 5, defined as $x$, satisfies the condition $x^2-x\geq 0$.
+    
+    $$\exists x\in\mathbb R\geq 5,x^2-x\geq 0$$
+
+Quantifiers can also be negated and nested. The opposite of "for each ... that satisfies $P(x)$" is "there exists ... that does **not** satisfy $P(x)$".
+
+$$\neg(\forall x\in S,P(x))\equiv\exists x\in S,\neg P(x)$$
+
+Nested quantifiers are **evaluated in sequence**. If the quantifiers are the same, they can be grouped together per the commutative and/or associative laws.
+
+$$\forall x\in\mathbb R,\forall y\in\mathbb R\equiv \forall x,y\in\mathbb R$$
+
+!!! warning
+    This means that the order of the quantifiers is relevant if the quantifiers are different:
+    
+    $\forall x\in\mathbb R,\exists y\in\mathbb R,x-y=1$ is **true** as setting $y$ to $x-1$ always fulfills the condition.
+    
+    $\exists y\in\mathbb R,\forall x\in\mathbb R, x-y=1$ is **false** as when $x$ is selected first, it is impossible for every value of $y$ to satisfy the open sentence.
+