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@@ -182,3 +182,39 @@ when `p` is true, `┓p` is false, and vice versa
 - Satisfied when either `p` and `q` are both true or both false
 
 - `p<->q` is "equivalent" to `(p->q)^(q->p)`
+
+---
+
+# 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`
+
+---
+
+[1.3 ->](1.3.md)
diff --git a/1.3.md b/1.3.md
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@@ -0,0 +1,101 @@
+[\<- 1.1](1.1.md)
+
+---
+
+# 1.3 Propositional Equivalences p1
+
+- Definition: We say propositinos `p` and `q` are **logically equivalent** if they have the **same truth table outputs**
+	- Notated as `p≡q` (three lign equal sign)
+	- kind of similar to `<->`
+
+- Defining terms using the conditional `p->q`
+	- **Converse**: `q->p` (if q, then p)
+	- **Inverse**: `┓p->┓q`
+	- **Contrapositive**: `┓q->┓p` (if not q, then not p)
+	- The original conditional and the contrapositive are equivalent
+		- So are the converse and the inverse
+	- Why is `p->q≡┓q->┓p`
+Reason 1: Truth Table
+
+|p|q|p->q|┓p|┓q|┓q->┓p|
+|-|-|----|--|--|------|
+|T|T|T   |F |F |T
+|T|F|F   |T |F |F
+|F|T|T   |F |T |T
+|F|F|T   |T |T |T
+
+- Since their truth table outputs are the same, these two statements are the same
+
+Reason 2: Logic/Venn Diagram
+
+- `p->q` -> "If we are in p, then we are in q"
+- `┓q->┓p` -> "If we are not in q, then we are not in p"
+- Both these statements describe the same venn diagram
+
+---
+
+- Ex: Show `(p->q)^(q->p)≡p<->q
+
+Truth Table:
+
+|p|q|p->q|q->p|(p->q)^(q->p)|p<->q|
+|-|-|----|----|-------------|-----|
+|T|T|T   |T   |T            |T    |
+|T|F|F   |T   |F            |F    |
+|F|T|T   |F   |F            |F    |
+|F|F|T   |T   |T            |T    |
+
+- The output is logically equivalent since their truth table outputs are the same
+
+- Ex: Fill in the blank:  ┓(I'm tall and I have black hair) ≡ I'm not tall _____ I don't have black hair
+	- The negation is saying, "it is not the case that..."
+	- The correct answer is `∨` (or)
+	
+- Now use a truth table to show `┓(p^q)≡┓p∨┓q`
+
+Truth Table:
+
+|p|q|┓(p^q)|┓p∨┓q|
+|-|-|------|-----|
+|T|T|F     |F    |
+|T|F|T     |T    |
+|F|T|T     |T    |
+|F|F|T     |T    |
+
+- This example is true because of DeMorgan's Law
+
+- Ex: Use a truth table to show that `p->q≡┓p∨q`
+
+Truth Table:
+
+|p|q|p->q|┓p∨q|
+|-|-|----|----|
+|T|T|T   |T   |
+|T|F|F   |F   |
+|F|T|T   |T   |
+|F|F|T   |T   |
+
+---
+
+# 1.3 Propositional Equivalences p2
+
+## Definitions
+- A proposition that is **always true** is a **tautology**
+	- (tautology≡`T`)
+	- e.g. `p∨┓p≡T`
+
+- A proposition that is **always false** is a **contradiction**
+	- (contradiction≡`F`)
+	- e.g. `p^┓p≡F`
+
+- A proposition that can be **true or false** is a **contingency**
+
+[Table of Common Equivalences](1.3.pdf)
+
+### Ways to show p≡q
+
+1. Truth Tables
+2. Show `p<->q` is a tautology (multiple options)
+	- with a truth table
+	- by (3)
+3. Use table of Common Equivalences (see link above)
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