199 lines
6.1 KiB
Markdown
199 lines
6.1 KiB
Markdown
# ECE 108: Discrete Math 1
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An **axiom** is a defined core assumption held to be true.
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!!! example
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True is not false.
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A **theorem** is a true statement derived from axioms via logic or other theorems.
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!!! example
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True or false is true.
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A **proposition/statement** must be able to have the property that it is exclusively true or false.
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!!! example
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The square root of 2 is a rational number.
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An **open sentence** becomes a proposition if a value is assigned to the variable.
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!!! example
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$x^2-x\geq 0$
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## Truth tables
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A truth table lists all possible **truth values** of a proposition, containing independent **statement variables**.
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!!! example
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| p | q | p and q |
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| --- | --- | --- |
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| T | T | T |
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| T | F | F |
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| F | T | F |
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| F | F | F |
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## Logical operators
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!!! definition
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- A **compound statement** is composed of **component statements** joined by logical operators AND and OR.
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The **negation** operator is equivalent to logical **NOT**.
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$$\neg p$$
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The **conjunction** operaetor is equivalent to logical **AND**.
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$$p\wedge q$$
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The **disjunction** operator is equivalent to logical **OR**.
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$$p\vee q$$
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### Proposation relations
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!!! definition
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A **tautology** is a statement that is always true, regardless of its statement variables.
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The **implication** sign requires that if $p$ is true, $q$ is true, such that *$p$ implies $q$*. The first symbol is the **hypothesis** and the second symbol is the **conclusion**.
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$$p\implies q$$
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| $p$ | $q$ | $p\implies q$ |
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| --- | --- | --- |
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| T | T | T |
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| T | F | F |
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| F | T | T |
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| F | F | F |
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The **inference** sign represents the inverse of the implication sign, such that $p$ **is implied by** $q$. It is equivalent to $q\implies p$.
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$$p\impliedby q$$
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The **if and only if** sign requires that the two propositions imply each other — i.e., that the state of $p$ is the same as the state of $q$. It is equivalent to $(p\implies q)\wedge (p\impliedby q)$.
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$$p\iff q$$
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The **logical equivalence** sign represents if the truth values for both statements are **the same for all possible variables**, such that the two are **equivalent statements**.
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$$p\equiv q$$
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$p\equiv q$ can also be defined as true when $p\iff q$ is a tautology.
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!!! warning
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$p\equiv q$ is *not a proposition* itself but instead *describes* propositions. $p\iff q$ is the propositional equivalent.
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## Common theorems
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The **double negation rule** states that if $p$ is a proposition:
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$$\neg(\neg p)\equiv p$$
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!!! tip "Proof"
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Note that:
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| $p$ | $\neg p$ | $\neg(\neg p)$ |
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| --- | --- | --- |
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| T | F | T |
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| F | T | F |
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Because the truth values of $p$ and $\neg(\neg p)$ for all possible truth values are equal, by definition, it follows that $p\equiv\neg(\neg p)$.
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!!! warning
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Proofs must include the definition of what is being proven, and any relevant evidence must be used to describe why.
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The two **De Morgan's Laws** allow distributing the negation operator in a dis/conjunction if the junction is inverted.
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$$
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\neg(p\vee q)\equiv(\neg p)\wedge(\neg q) \\
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\neg(p\wedge q)\equiv(\neg p)\vee(\neg q)
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$$
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An implication can be expressed as a disjunction. As long as it is stated, it can used as its definition.
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$$p\implies \equiv (\neg p)\vee q$$
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Two **converse** propositions imply each other:
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$$p\implies q\text{ is the converse of }q\implies p$$
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A **contrapositive** is the negatated converse, and is **logically equivalent to the original implication**. This allows proof by contrapositive.
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$$\neg p\implies\neg q\text{ is the contrapositive of }q\implies p$$
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### Operator laws
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Both **AND** and **OR** are commutative.
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$$
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p\wedge q\equiv q\wedge p \\
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p\vee q\equiv q\vee p
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$$
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Both **AND** and **OR** are associative.
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$$
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(p\wedge q)\wedge r\equiv p\wedge(q\wedge r) \\
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(p\vee q)\vee r\equiv p\vee(q\vee r)
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$$
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Both **AND** and **OR** are distributive with one another.
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$$
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p\wedge(q\vee r)\equiv(p\wedge q)\vee(p\wedge r) \\
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p\vee(q\wedge r)\equiv(p\vee q)\wedge(p\vee r)
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$$
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!!! tip "Proof"
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$$
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\begin{align*}
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(\neg p\vee\neg r)\wedge s\wedge\neg t&\equiv\neg(p\wedge r\vee s\implies t) \\
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\tag*{definition of implication} &\equiv \neg (p\wedge r\vee[\neg s\vee t]) \\
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\tag*{DML} &\equiv\neg(p\wedge r)\wedge\neg[(\neg s)\vee t)] \\
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\tag*{DML} &\equiv(\neg p\vee\neg r)\wedge\neg[(\neg t)\vee t] \\
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\tag*{DML} &\equiv(\neg p\vee\neg r)\wedge\neg(\neg s)\wedge\neg t \\
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\tag*{double negation} &\equiv(\neg p\vee\neg r)\wedge s\wedge\neg t
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\end{align*}
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$$
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### Quantifiers
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A **quantified statement** includes a **quantifier**, **variable**, **domain**, and **open sentence**.
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$$
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\underbrace{\text{for all}}_\text{quantifier}\ \underbrace{\text{real numbers}\overbrace{x}^\text{variable}\geq 5}_\text{domain}, \underbrace{x^2-x\geq 0}_\text{open sentence}
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$$
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The **universal quantifier** $\forall$ indicates "for all".
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$$\forall x\in S,P(x)$$
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!!! example
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All real numbers greater than or equal to 5, defined as $x$, satisfy the condition $x^2-x\geq 0$.
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$$\forall x\in\mathbb R\geq 5,x^2-x\geq 0$$
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The **existential quantifier** $\exists$ indicates "there exists at least one".
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$$\exists x\in S, P(x)$$
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!!! example
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There exists at least one real number greater than or equal to 5, defined as $x$, satisfies the condition $x^2-x\geq 0$.
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$$\exists x\in\mathbb R\geq 5,x^2-x\geq 0$$
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Quantifiers can also be negated and nested. The opposite of "for each ... that satisfies $P(x)$" is "there exists ... that does **not** satisfy $P(x)$".
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$$\neg(\forall x\in S,P(x))\equiv\exists x\in S,\neg P(x)$$
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Nested quantifiers are **evaluated in sequence**. If the quantifiers are the same, they can be grouped together per the commutative and/or associative laws.
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$$\forall x\in\mathbb R,\forall y\in\mathbb R\equiv \forall x,y\in\mathbb R$$
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!!! warning
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This means that the order of the quantifiers is relevant if the quantifiers are different:
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$\forall x\in\mathbb R,\exists y\in\mathbb R,x-y=1$ is **true** as setting $y$ to $x-1$ always fulfills the condition.
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$\exists y\in\mathbb R,\forall x\in\mathbb R, x-y=1$ is **false** as when $x$ is selected first, it is impossible for every value of $y$ to satisfy the open sentence.
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