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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) |