122 lines
3.5 KiB
Markdown
122 lines
3.5 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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