# ECE 108: Discrete Math 1

An **axiom** is a defined core assumption held to be true.

!!! example
    True is not false.

A **theorem** is a true statement derived from axioms via logic or other theorems.

!!! example
    True or false is true.

A **proposition/statement** must be able to have the property that it is exclusively true or false.

!!! example
    The square root of 2 is a rational number.

An **open sentence** becomes a proposition if a value is assigned to the variable.

!!! example
    $x^2-x\geq 0$

## Truth tables

A truth table lists all possible **truth values** of a proposition, containing independent **statement variables**.

!!! example
    | p | q | p and q |
    | --- | --- | --- |
    | T | T | T |
    | T | F | F |
    | F | T | F |
    | F | F | F |

## Logical operators

!!! definition
    - A **compound statement** is composed of **component statements** joined by logical operators AND and OR.

The **negation** operator is equivalent to logical **NOT**.

$$\neg p$$

The **conjunction** operaetor is equivalent to logical **AND**.

$$p\wedge q$$

The **disjunction** operator is equivalent to logical **OR**.

$$p\vee q$$

### Proposation relations

!!! definition
    A **tautology** is a statement that is always true, regardless of its statement variables.

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**.

$$p\implies q$$

| $p$ | $q$ | $p\implies q$ |
| --- | --- | --- |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | F |

The **inference** sign represents the inverse of the implication sign, such that $p$ **is implied by** $q$. It is equivalent to $q\implies p$.

$$p\impliedby q$$

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 $(p\implies q)\wedge (p\impliedby q)$.

$$p\iff q$$

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**.

$$p\equiv q$$

$p\equiv q$ can also be defined as true when $p\iff q$ is a tautology.

!!! warning
    $p\equiv q$ is *not a proposition* itself but instead *describes* propositions. $p\iff q$ is the propositional equivalent.

## Common theorems

The **double negation rule** states that if $p$ is a proposition:

$$\neg(\neg p)\equiv p$$

!!! tip "Proof"
    Note that:
    | $p$ | $\neg p$ | $\neg(\neg p)$
    | --- | --- | --- |
    | T | F | T |
    | F | T | F |
    
    Because the truth values of $p$ and $\neg(\neg p)$ for all possible truth values are equal, by definition, it follows that $p\equiv\neg(\neg p)$.

!!! warning
    Proofs must include the definition of what is being proven, and any relevant evidence must be used to describe why.

The two **De Morgan's Laws** allow distributing the negation operator in a dis/conjunction if the junction is inverted.

$$
\neg(p\vee q)\equiv(\neg p)\wedge(\neg q) \\
\neg(p\wedge q)\equiv(\neg p)\vee(\neg q)
$$

An implication can be expressed as a disjunction. As long as it is stated, it can used as its definition.

$$p\implies \equiv (\neg p)\vee q$$