# Propositional Logic p1

- Def: A **proposiion** is a declarative sentence that is either true or false (but not both)
	- A sentence that declares a fact

- Determine which of the following sentences are propositions:
	1. Lebron James plays basketball
		- it is declaring something
		- it is true
		- **it is a proposition** :)
	2. All students live in dorms
		- it is declaring something
		- it is false
		- **it is a proposition** :)
	3. Where is Carmen?
		- it is NOT declaring something
		- **it is NOT a proposition** :(
	4. Sit down!
		- it is NOT declaring something
		- **it is NOT a proposition** :(
	5. `4 + 5 = 9`
		- it is declaring something
		- it is true
		- **it is a proposition** :)
	6. `x + 5 = 9`
		- it is declaring something
		- **it is NEITHER true or false**
		- **it is NOT a proposition** :(
	7. `x + 5y = 5y + x; x,y are real`
		- it is declaring something
		- it is true
		- **it is a proposition** :)

- If a proposition is true: write `T`
- If a proposition is false: write `F`
	- alternatively, sometimes an upside down 'T' instead

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# Operators

- List of operators:
	- ┓ ([a top right corner](negation.png))
	- ^ (carrot)
	- ∨ (descending wedge symbol)
	- ⊕ ([plus inside a circle](plus_inside_circle.png))
	- -> (right arrow)
	- <-> (arrow pointed both ways)
- These are all **operators** on propositions
	- Propositions involving operators are **compound propositions**
		- (e.g. p^q)
			- read as *p and q*

## Negation (┓)

- Let `p` be a proposition `┓p` (meaning *not p*)
	- This is a **negation**
	- This is a **proposition**

- Let `r` = "Lebron James plays basketball"
	- `┓r` = "Lebron James does **not play** basketball"

## Introducing Truth Tables

example:

|p|┓p|
|-|--|
|T|F |
|F|T |

when `p` is true, `┓p` is false, and vice versa

## Conjunction (^)

- Let `p` and `q` be propositions
- "p and q" is a proposition called the **conjunction** of `p` and `q`, denoted by `p^q`
	- read as "p and q"

### Truth Table

|p|q|p^q|
|-|-|---|
|T|T|T  |
|T|F|F  |
|F|T|F  |
|F|F|F  |

`p^q` is only true when both `p` and `q` are true

## Disjunction (∨)

- Let `p` and `q` be propositions
- "p or q" is a proposition called the **disjunction** of `p` and `q`, denoted `p∨q`
	- read as "p or q"

### Truth Table

|p|q|p∨q|
|-|-|---|
|T|T|T  |
|T|F|T  |
|F|T|T  |
|F|F|F  |

- The disjunction is an **inclusive or**
	- Either `p`, or `q`, or both need to be true for `p∨q` to be true

## Exclusive Or (⊕)

- Let `p` and `q` be propositions
- "p exclusive or q" is a proposition called **exclusive or** of `p` and `q` and is denoted `p⊕q`
	- read as "p exclusive or q"

### Truth Table

|p|q|p⊕q|
|-|-|---|
|T|T|F  |
|T|F|T  |
|F|T|T  |
|F|F|F  |

- One of `p` or `q` need to be true for `p⊕q` to be true, but not both

## "Or" in English

- I can wake up early **or** I can sleep in.
	- In English, this is an exclusive or (I can't wake up and sleep at the same time!)

- People with kids **or** pets get less sleep.
	- In English, this is an inclusive or (both kids and pets can get less sleep)

## Memory Tip
- `^` looks like an intersection symbol (which means **and**)
- `∨` looks like an union symbol (which means **inclusive or**)

## Implication (->)

- Sometimes called conditional operator
- Let `p` and `q` be propositions
- "If p then q" is a proposition called the **implication** of `p` and `q` and is denoted `p->q`
	- read as "if p then q" or sometimes "p implies q"

### Truth Table

|p|q|p->q|
|-|-|----|
|T|T|T   |
|T|F|F   |
|F|T|T   |
|F|F|T   |

- Example: "If it is Wednesday, we wear pink"
	- `p` is "it is Wednesday" (sometimes called the hypothesis)
	- `q` is "we wear pink" (sometimes called the conclusion)
	- Only violated (`p->q` is false) if it is Wednesday (`p` is true) and we don't wear pink (`q` is false)

- Implication can be read in many different ways (example `p->q`):
	- p implies q
	- p only if q
	- q when p
	- p if sufficient for q
	- q is necessary for p

## Biconditional (<->)

- Let `p` and `q` be propositions
- "p if and only if q" is a proposition called the **biconditional** of `p` and `q`, denoted `p<->q`
	- read "p if and only if q" or "p iff q"

### Truth Table

|p|q|p<->q|
|-|-|----|
|T|T|T   |
|T|F|F   |
|F|T|F   |
|F|F|T   |

- Violated when one is `T` but other if `F`
- Satisfied when either `p` and `q` are both true or both false

- `p<->q` is "equivalent" to `(p->q)^(q->p)`

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# Propositional Logic p2

- ex: What operator do we do first in `┓p^q<->r`

## Precedence (order of operations)

1. `┓`
2. `^`, `∨`, `⊕`
	- "Exclusive or" (`⊕`) is sometimes referred to as "Ex Or" for short
	- Need parenthesis to differentiate among these
3. `->`, `<->`
	- Need parenthesis to differentiate among these
	
`┓p^q<->r` means `(┓p^q)<->r`

- ex: `p^q∨r`
	- is ambiguous (not sure what to do first) because of lack of parethesis

- ex: Translate the following sentence to logic:
	- "You can be Draco's friend **only if** you are in Slytherin **or** you eat floating cupcakes"
	- let `d` = "You can be Draco's friend
	- let `s` = "You are in Slytherin"
	- let `c` = "You eat floating cupcakes"
	- `d->s∨c` or `d->(s∨c)`
		- Mean the same thing

- Note: "p if q" means `p<-q` (or `q->p`)
- Note: "p only if q" means `p->q`
- Note: "p if and only if q" means `p<->q`

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[1.3 ->](1.3.md)